Vector Algebra

Q34 The vector $\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}}$ bisects the angle between the non-collinear vectors $\overrightarrow {\rm{a}}$ and $\overrightarrow {\rm{b}} ,$ if............. FillBlank Q35 If $\overrightarrow {\rm{r}} \cdot \overrightarrow {\rm{a}} = 0,$ $\overrightarrow {\rm{r}} \cdot \overrightarrow {\rm{b}} = 0$ and $\overrightarrow {\rm{r}} \cdot \overrightarrow {\rm{c}} = 0$ for some non-zero vector $\overrightarrow {\rm{r}} ,$ then the value of $\overrightarrow {\rm{a}} \cdot (\overrightarrow {\rm{b}} \times \overrightarrow {\rm{c}} )$ is.............. FillBlank Q36 The vectors $\overrightarrow {\rm{a}} = 3\widehat {\rm{i}} - 2\widehat {\rm{j}} + 2$and $\overrightarrow {\rm{b}} = - \widehat {\rm{i}} - 2\widehat {\rm{k}}$ are the adjacent sides of

a parallelogram. The angle between its diagonals is...... FillBlank Q37 The values of $k$, for which $|k\overrightarrow {\rm{a}} | < \overrightarrow {\rm{a}} \mid$ and $k\overrightarrow {\rm{a}} + \frac{1}{2}\overrightarrow {\rm{a}}$ is parallel to $\overrightarrow {\rm{a}}$ holds true are …………… FillBlank Q38 The value of the expression $|\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} {|^2} + {(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} )^2}$ is………. FillBlank Q39 If $|\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} {|^2} + |\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} {|^2} = 144$ and $|\overrightarrow {\rm{a}} | = 4,$ then $|\overrightarrow {\rm{b}} |$ is equal to FillBlank Q40 If $\overrightarrow {\rm{a}}$ is any non-zero vector, then $(\overrightarrow {\rm{a}} \cdot \widehat {\rm{i}}) \cdot \widehat {\rm{i}} + (\overrightarrow {\rm{a}} \cdot \widehat {\rm{j}}) \cdot \widehat {\rm{j}} + (\overrightarrow {\rm{a}} \cdot \widehat {\rm{k}})\widehat {\rm{k}}$ is equal
to ………… FillBlank Q15 Prove that in any $\Delta ABC,$ $\cos A = \frac{{{b^2} + {c^2} - {a^2}}}{{2bc}}$, where $a,b$ and $c$ are the magnitudes of the sides opposite to the vertices $A,B$ and $C$, respectively.
Here, components of $C$ are $c\cos A$ and $c\sin A$ is drawn. LA Q16 If $\overrightarrow {\rm{a}} ,\overrightarrow {\rm{b}}$ and $\overrightarrow {\rm{c}}$ determine the vertices of a triangle, show that $\frac{1}{2}[\overrightarrow {\rm{b}} \times \overrightarrow {\rm{c}} + \overrightarrow {\rm{c}} \times \overrightarrow {\rm{a}} + \overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} ]$ gives the vector area of the triangle. Hence, deduce the condition that the three points $\overrightarrow {\rm{a}} ,\overrightarrow {\rm{b}}$ and $\overrightarrow {\rm{c}}$ are collinear. Also, find the unit vector normal to the plane of the triangle. LA Q17 Show that area of the parallelogram whose diagonals are given by $\overrightarrow {\rm{a}}$ and $\overrightarrow {\rm{b}}$ is $\frac{{|\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} |}}{2}$. Also, find the area of the parallelogram, whose diagonals are $2\widehat {\rm{i}} - \widehat {\rm{j}} + k$ and $\widehat {\rm{i}} + 3\widehat {\rm{j}} - \widehat {\rm{k}}$. LA Q18 If $\overrightarrow {\rm{a}} = \widehat {\rm{i}} - \widehat {\rm{j}} + \widehat {\rm{k}}$ and $\overrightarrow {\rm{b}} = \widehat {\rm{j}} - \widehat {\rm{k}},$ then find a vector $\overrightarrow {\rm{c}}$ such that $\overrightarrow {\rm{a}} \times \overrightarrow {\rm{c}} = \overrightarrow {\rm{b}}$ and $\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{c}} = 3$. LA Q19 The vector in the direction of the vector $\widehat {\rm{i}} - 2\widehat {\rm{j}} + 2\widehat {\rm{k}}$ that has magnitude 9 is MCQ Q20 The position vector of the point which divides the join of points $2\overrightarrow {\rm{a}} - 3\overrightarrow {\rm{b}}$ and $\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}}$ in the ratio 3: 1, is MCQ Q21 The vector having initial and terminal points as (2,5,0) and (-3,7, 4), respectively is MCQ Q22 The angle between two vectors $\overrightarrow {\rm{a}}$ and $\overrightarrow {\rm{b}}$ with magnitudes $\sqrt 3$ and 4 respectively and $\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} = 2\sqrt 3$ is MCQ Q23 Find the value of $\lambda$ such that the vectors $\overrightarrow {\rm{a}} = 2\widehat {\rm{i}} + \lambda \widehat {\rm{j}} + \widehat {\rm{k}}$ and $\overrightarrow {\rm{b}} = \widehat {\rm{i}} + 2\widehat {\rm{j}} + 3\widehat {\rm{k}}$ are orthogonal. MCQ Q24 The value of $\lambda$ for which the vectors $3\widehat {\rm{i}} - 6\widehat {\rm{j}} + \widehat {\rm{k}}$ and $2\widehat {\rm{i}} - 4\widehat {\rm{j}} + \lambda \widehat {\rm{k}}$ are parallel, is MCQ Q25 The vectors from origin to the points $A$ and $B$ are $\overrightarrow {\rm{a}} = 2\widehat {\rm{i}} - 3\widehat {\rm{j}} + 2\widehat {\rm{k}}$ and $\overrightarrow {\rm{b}} = 2\widehat {\rm{i}} + 3\widehat {\rm{j}} + \widehat {\rm{k}}$ respectively, then the area of $\Delta OAB$ is equal to MCQ Q26 For any vector $\overrightarrow {\rm{a}}$, the value of ${(\overrightarrow {\rm{a}} \times \widehat {\rm{i}})^2} + {(\overrightarrow {\rm{a}} \times \widehat {\rm{j}})^2} + {(\overrightarrow {\rm{a}} \times \widehat {\rm{k}})^2}$ is MCQ Q27 If $|\overrightarrow {\rm{a}} | = 10,$ $|\overrightarrow {\rm{b}} | = 2$ and $\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} = 12$, then the value of $|\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} |$ is MCQ Q28 The vectors $\lambda \widehat {\rm{i}} + \widehat {\rm{j}} + 2\widehat {\rm{k}},$ $\widehat {\rm{i}} + \lambda \widehat {\rm{j}} - \widehat {\rm{k}}$ and $2\widehat {\rm{i}} - \widehat {\rm{j}} + \lambda \widehat {\rm{k}}$ are coplanar, if MCQ Q29 If $\overrightarrow {\rm{a}} ,\overrightarrow {\rm{b}}$ and $\overrightarrow {\rm{c}}$ are unit vectors such that $\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}} + \overrightarrow {\rm{c}} = \vec 0$, then the value ${\mathop{\rm of}\nolimits} \overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} + \overrightarrow {\rm{b}} \cdot \overrightarrow {\rm{c}} + \overrightarrow {\rm{c}} \cdot \overrightarrow {\rm{a}}$ is MCQ Q30 The projection vector of $\overrightarrow {\rm{a}}$ on $\overrightarrow {\rm{b}}$ is MCQ Q31 If $\overrightarrow {\rm{a}} ,\overrightarrow {\rm{b}}$ and $\overrightarrow {\rm{c}}$ are three vectors such that $\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}} + \overrightarrow {\rm{c}} = \vec 0$ and $|\overrightarrow {\rm{a}} | = 2$,$|\overrightarrow {\rm{b}} | = 3$ and $|\overrightarrow {\rm{c}} | = 5$, then the value of $\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} + \overrightarrow {\rm{b}} \cdot \overrightarrow {\rm{c}} + \overrightarrow {\rm{c}} \cdot \overrightarrow {\rm{a}}$ is MCQ Q32 If $|\overrightarrow {\rm{a}} | = 4$ and $- 3 \le \lambda \le 2$, then the range of $|\lambda \overrightarrow {\rm{a}} |$ is MCQ Q33 The number of vectors of unit length perpendicular to the vectors $\overrightarrow {\rm{a}} = 2\widehat {\rm{i}} + \widehat {\rm{j}} + 2\widehat {\rm{k}}$ and $\overrightarrow {\rm{b}} = \widehat {\rm{j}} + \widehat {\rm{k}}$ is MCQ Q1 Find the unit vector in the direction of sum of vectors $\overrightarrow {\rm{a}} = 2\widehat {\rm{i}} - \widehat {\rm{j}} + \widehat {\rm{k}}$ and $\overrightarrow {\rm{b}} = 2\widehat {\rm{j}} + \widehat {\rm{k}}$. SA Q2 If $\overrightarrow {\rm{a}} = \widehat {\rm{i}} + \widehat {\rm{j}} + 2\widehat {\rm{k}}$ and $\overrightarrow {\rm{b}} = 2\widehat {\rm{i}} + \widehat {\rm{j}} + 2\widehat {\rm{k}}$, then find the unit vector in the direction of

(i) $6\overrightarrow {\rm{b}}$

(ii) $2\overrightarrow {\rm{a}} - \overrightarrow {\rm{b}}$ SA Q3 Find a unit vector in the direction of $\overrightarrow {P{\rm{Q}}}$, where $P$ and $Q$ have coordinates (5,0,8) and $(3,3,2)$, respectively.
Since, the coordinates of $P$ and $Q$ are (5,0,8) and (3,3,2) , respectively. SA Q4 If $\overrightarrow {\rm{a}}$ and $\overrightarrow {\rm{b}}$ are the position vectors of $\overrightarrow {\rm{A}}$ and $\overrightarrow {\rm{B}}$ respectively, then find the position vector of a point $\overrightarrow {\rm{C}}$ in $\overrightarrow {{\rm{BA}}}$ produced such that $\overrightarrow {{\rm{BC}}} = 1.5\overrightarrow {{\rm{BA}}}$. SA Q5 Using vectors, find the value of $k$, such that the points $(k, - 10,3)$, (1,-1,3) and (3,5,3) are collinear. SA Q6 A vector $\overrightarrow {\rm{r}}$ is inclined at equal angles to the three axes. If the magnitude of $\overrightarrow {\rm{r}}$ is $2\sqrt 3$ units, then find the value of $\overrightarrow {\rm{r}}$. SA Q7 If a vector $\overrightarrow {\rm{r}}$ has magnitude 14 and direction ratios 2,3 and -6 . Then, find the direction cosines and components of $\overrightarrow {\rm{r}}$, given that $\overrightarrow {\rm{r}}$ makes an acute angle with X-axis. SA Q8 Find a vector of magnitude $6$, which is perpendicular to both the vectors $2\widehat {\rm{i}} - \widehat {\rm{j}} + 2\widehat {\rm{k}}$ and $4\widehat {\rm{i}} - \widehat {\rm{j}} + 3\widehat {\rm{k}}$. SA Q9 Find the angle between the vectors $2\widehat {\rm{i}} - \widehat {\rm{j}} + \widehat {\rm{k}}$ and $3\widehat {\rm{i}} + 4\widehat {\rm{j}} - \widehat {\rm{k}}$. SA Q10 If $\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}} + \overrightarrow {\rm{c}} = 0$, then show that $\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} = \overrightarrow {\rm{b}} \times \overrightarrow {\rm{c}} = \overrightarrow {\rm{c}} \times \overrightarrow {\rm{a}}$. Interpret the result geometrically. SA Q11 Find the sine of the angle between the vectors $\overrightarrow {\rm{a}} = 3\widehat {\rm{i}} + \widehat {\rm{j}} + 2\widehat {\rm{k}}$ and $\overrightarrow {\rm{b}} = 2\widehat {\rm{i}} - 2\widehat {\rm{j}} + 4\widehat {\rm{k}}$. SA Q12 If $A,B,C$ and $D$ are the points with position vectors $\widehat {\rm{i}} - \widehat {\rm{j}} + \widehat {\rm{k}}$, $2\widehat {\rm{i}} - \widehat {\rm{j}} + 3\widehat {\rm{k}},$ $2\widehat {\rm{i}} - 3\widehat {\rm{k}}$ and $3\widehat {\rm{i}} - 2\widehat {\rm{j}} + \widehat {\rm{k}}$ respectively, then find the projection of $\overrightarrow {{\rm{AB}}}$ along $\overrightarrow {{\rm{CD}}}$. SA Q13 Using vectors, find the area of the $\Delta ABC$ with vertices $A(1,2,3)$, $B(2, - 1,4)$ and $C(4,5, - 1)$ SA Q14 Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.
Let $ABCD$ and $ABFE$ are parallelograms on the same base $AB$ and between the same parallel lines $AB$ and $DF$. SA

Q1 Represent graphically a displacement of $40$ km ${30^o}$ east of north.
SA Q3 Classify the following as scalar and vector quantities.

(i) time period

(ii) distance

(iii) force

(iv) velocity

(v) work done SA Q4 In the given figure (a square), identify the following vectors.

(i) Coinitial

(ii) Equal

(iii) Collinear but not equal

SA Q5 Answer the following as true or false.

(i) $\vec a$ and$- \vec a$ are collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal. SA

Q1 Compute the magnitude of the following vectors:
$\vec a = \hat i + \hat j + \hat k;\,\,\,\vec b = 2\hat i - 7\hat j - 3\hat k;\,\,\vec c = \cfrac{1}{{\sqrt 3 }}\hat i + \cfrac{1}{{\sqrt 3 }}\hat j - \cfrac{1}{{\sqrt 3 }}\hat k$
SA Q2 Write two different vectors having same magnitude. SA Q3 . Write two different vectors having same direction. SA Q4 Find the values of $x$ and $y$ so that the vectors $2\hat i + 3\hat j$ and $x\hat i + y{\hat j^:}$ are equal. SA Q5 Find the scalar and vector components of the vector with initial point $(2,1)$ and terminal point $( - 5,7)$. SA Q6 Find the sum of the vectors
$\vec a = \hat i - 2\hat j + \hat k,\vec b = - 2\hat i + 4\hat j + 5\hat k$ and $\vec c = \hat i - 6\hat j - 7\hat k$. SA Q7 Find the unit vector in the direction of the vector
$\vec a = \hat i + \hat j + 2\hat k$.
SA Q8 Find the unit vector in the direction of vector $\overrightarrow {PQ}$, where $P$ and $Q$ are the points $(1,2,3)$ and $(4,5,6)$ respectively.
SA Q9 For given vectors, $\vec a = 2\hat i - \hat j + 2\hat k$ and $\vec b = - \hat i + \hat j - \hat k$,

find the unit vector in the direction of the vector $\vec a + \vec b$. SA Q10 Find a vector in the direction of vector $5\hat i - \hat j + 2\hat k$ which has magnitude $8$ units. SA Q11 Show that the vectors $2\hat i - 3\hat j + 4\hat k$ and $- 4\hat i + 6\hat j - 8\hat k$ are collinear. SA Q12 Find the direction cosines of the vector $\hat i + 2\hat j + 3\hat k$.
SA Q13 Find the direction cosines of the vector joining the points $A(1,2, - 3)$ and $B( - 1, - 2,1)$ directed from $A$ to $B$. SA Q14 Show that the vector $\hat i + \hat j + \hat k$ is equally inclined to the axes $OX,OY$ and $OZ$.
SA Q15 Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $\hat i + 2\hat j - \hat k$ and $- \hat i + \hat j + \hat k$ respectively, in the ratio 2 : 1

(i) internally

(ii) externally SA Q16 Find the position vector of the mid point of the vector joining the points $P(2,3,4)\;{\rm{and}}\;Q(4,1,\; - 2)$ .
SA Q17 Show that the points $A,B$ and $C$ with position vectors, $\vec a = 3\hat i - 4\hat j - 4\hat k,\,\,\,\vec b = 2\hat i - \hat j + \hat k$ and $\vec c = \hat i - 3\hat j - 5\hat k$, respectively, form the vertices of a right angled triangle.
SA Q18 In triangle $ABC$, which of the following is not true:

• $\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CA} = \vec 0$

• $\overrightarrow {AB} + \overrightarrow {BC} - \overrightarrow {AC} = \vec 0$

• $\overrightarrow {AB} + \overrightarrow {BC} - \overrightarrow {CA} = \vec 0$

• $\overrightarrow {AB} - \overrightarrow {CB} + \overrightarrow {CA} = \vec 0$

SA Q19 If $\vec a$ and $\vec b$ are two collinear vectors, then which of the following is incorrect:

• $\vec b = \lambda \vec a$, for some scalar $\lambda$

• $\vec a = \pm \,\,\vec b$

• the respective components of $\vec a$ and $\vec b$ are proportional

• both the vectors $\vec a$ and $\vec b$ have same direction, but different magnitudes.

SA

Q Find the projection of the vector $\hat i + 3\hat j + 7\hat k$ on the vector $7\hat i - \hat j + 8\hat k$
447,Ex.10.3,Q.No 4] SA Q1 Find the angle between two vectors $\vec a$ and $\vec b$ with magnitudes $\sqrt 3$ and $2$ respectively, having $\vec a \cdot \vec b = \sqrt 6$.
SA Q2 Find the angle between the vectors $\hat i. - 2\hat j + 3\hat k$and $3\hat i - 2\hat j + \hat k$ .
SA Q3 Find the projection of the vector $\hat i - \hat j$ on the vector $\hat i + \hat j$.
SA Q5 Show that each of the given three vectors is a unit vector: $\cfrac{1}{7}(2\hat i + 3\hat j + 6\hat k),\,\,\cfrac{1}{7}(3\hat i - 6\hat j + 2\hat k),\;\,\cfrac{1}{7}(6\hat i + 2\hat j - 3\hat k)$ .
Also, show that they are mutually perpendicular to each other. SA Q6 Find $|\vec a|$and$|\vec b|$, if$(\vec a + \vec b) \cdot (\vec a - \vec b) = 8$ and $|\vec a| = 8|\vec b|$. SA Q7 Evaluate the product $(3\vec a - 5\vec b) \cdot (2\vec a + 7\vec b)$.
SA Q8 Find the magnitude of two vectors $\vec a$ and $\vec b$ having the same magnitude and such that the angle between them is ${60^\circ }$ and their scalar product is $\cfrac{1}{2}$.
SA Q9 Find $|\vec x|$, if for a unit vector $\vec a,(\vec x - \vec a) \cdot (\vec x + \vec a) = 12$.
SA Q10 If $\vec a = 2\hat i + 2\hat j + 3\hat k,\,\,\vec b = - \hat i + 2\hat j + \hat k$ and $\vec c = 3\hat i + \hat j$ are such that $\vec a + \lambda \vec b$ is perpendicular to $\vec c$, then find the value of $\lambda$.
SA Q11 Show that $|\vec a|\vec b + |\vec b|\vec a$ is perpendicular to $|\vec a|\vec b - |\vec b|\vec a$, for any two non-zero vectors $\vec a$ and $\vec b$. SA Q12 If $\vec a \cdot \vec a = 0\,\,{\rm{and }}\vec a \cdot \vec b = 0$, then what can be concluded about the vector $\vec b$?
SA Q13 If $\vec a,\vec b,\vec c$ are unit vectors such that $\overrightarrow a + \vec b + \vec c = 0$, find the value of $\vec a \cdot \vec b + \vec b \cdot \vec c + \vec c \cdot \vec a$.
SA Q14 If either vector $\vec a = \vec 0$ or $\vec b = \vec 0$, then $\vec a \cdot \vec b = 0$. But the converse need not be true. Justify $yo1\pi$ answer with an example. SA Q15 If the vertices $A,B,C$ of a triangle $ABC$ are$(1,2,3),( - 1,0,0),(0,1,2)$ , respectively, then find $\angle ABC.[\angle ABC$ is the angle between the vectors $\overrightarrow {BA}$ and $\overrightarrow {BC}$].
SA Q16 Show that the points $A(1,2,7),\,\,\,B(2,6,3)$ and $C(3,10, - 1)$ are collinear. SA Q17 Show that the vectors $2\hat i - \hat j + \hat k,\hat i - 3\hat j - 5\hat k$ and $3\hat i - 4\hat j - 4\hat k$ form the vertices of a right angled triangle.
SA Q18 If $\vec a$ is a non-zero vector of magnitude $'a'$ and $\lambda$ a non- zero scalar, then $\lambda \vec a$ is unit vector if

• $\lambda = 1$

• $\lambda = - 1$

• $a = |\lambda |$

• $a = 1/|\lambda |$

SA

Q1 Find $|\vec a \times \vec b|$, if $\vec a = \hat i - 7\hat j + 7\hat k$ and $\vec b = 3\hat i - 2\hat j + 2\hat k$. SA Q2 Find a unit vector perpendicular to each of the vectors $\vec a + \vec b$ and $\vec a - \vec b$, where $\vec a = 3\hat i + 2\hat j + 2\hat k$ and $\vec b = \hat i + 2\hat j - 2\hat k$.
SA Q3 If a unit vector $\vec a$ makes angles $\cfrac{\pi }{3}$ with $\hat i,\cfrac{\pi }{4}$with $\hat j$ and an acute angle $\theta$ with $\hat k$ , then find $\theta$ and hence, the components of $\vec a$. SA Q4 Show that $(\vec a - \vec b) \times (\vec a + \vec b) = 2(\vec a \times \vec b)$
SA Q5 Find $\lambda$ and $\mu$ if $(2\hat i + 6\hat j + 27\hat k) \times (\hat i + \lambda \widehat j + \mu \hat k) = \vec 0$. SA Q6 Given that $\vec a \cdot \vec b = 0$ and $\vec a \times \vec b = \vec 0$. What can you conclude about the vectors $\vec a$ and $\vec b?$
SA Q7 Let the vectors $\vec a,\vec b,\vec c$ be given as ${a_1}\hat i + {a_2}\hat j + {a_3}\hat k,{b_1}\hat i + {b_2}\hat j + {b_3}\hat k,{c_1}\hat i + {c_2}\hat j + {c_3}\hat k$. Then show that $\vec a \times (\vec b + \vec c) = \vec a \times \vec b + \vec a \times \vec c$. SA Q8 If either $\vec a = \vec 0$ or $\vec b = \vec 0$, then $\vec a \times \vec b = \vec 0$. Is the converse true? Justify your answer with an example.
SA Q9 Find the area of the triangle with vertices $A(1,1,2)$ , $B(2,3,5)$ and $C(1,5,5)$.
SA Q10 Find the area of the parallelogram whose adjacent sides are determined by the vectors $\vec a = \hat i - \hat j + 3\hat k$ and $\vec b = 2\hat i - 7\hat j + \hat k$. SA Q11 Let the vectors $\vec a$ and $\vec b$ be such that $|\vec a| = 3$ and $|\vec b| = \cfrac{{\sqrt 2 }}{3}$, then $\vec a \times \vec b$ is a unit vector, if the angle between $\vec a$and $\vec b$ is

• $\pi /6$

• $\pi /4$

• $\pi /3$

• $\pi /2$

SA Q12 Area of a rectangle having vertices $A,B,C$ and $D$ with position vectors
$- \hat i + \cfrac{1}{2}\hat j + 4\hat k,\,\,\,\hat i + \cfrac{1}{2}\hat j + 4\hat k,\,\,\,\hat i - \cfrac{1}{2}\hat j + 4\hat k$ and $- \hat i - \cfrac{1}{2}\hat j + 4\hat k$ respectively is

• $\cfrac{1}{2}$

• $1$

• $2$

• $4$

SA

Q1 Write down a unit vector in $XY -$plane, making an angle of${30^o}$ with the positive direction of $x$-axis.
SA Q2 Find the scalar components and magnitude of the vector joining the points $P({x_1},\;{y_1},\;{z_1})\;and\;Q({x_2},\;{y_2},\;{x_2})$ . SA Q3 A girl walks $4$ km towards west, then she walks $3$ km in a direction ${30^o}$ east of north and stops. Determine the girl's displacement from her initial point of departure. SA Q4 If $\vec a = \vec b + \vec c$, then is it true that $|\vec a| = |\vec b| + |\vec c|$? Justify your answer. SA Q5 Find the value of $x$ for which $x(\hat i + \hat j + \hat k)$ is a unit vector.
SA Q6 Find a vector of magnitude 5 units, and parallel to the resultant of the vectors $\overrightarrow a = 2\hat i + 3\hat j - \hat k$ and $\overrightarrow b = \hat i - 2\hat j + \hat k$ . SA Q7 If $\overrightarrow a = \hat i + \widehat j + \hat k,\overrightarrow b = 2\hat i - \hat j + 3\hat k$ and $\overrightarrow c = \hat i - 2\widehat j + \hat k$,
find a unit vector parallel to the vector$2\overrightarrow a - \overrightarrow b + 3\overrightarrow c$.
SA Q8 Show that the points $A(1,\; - 2,\; - 8),\;B(5,0,\; - 2)$ and $C(11,3,7)$ are collinear and find the ratio in which $B$ divides $AC$. SA Q9 Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $(2\vec a + \vec b)$ and $(\overrightarrow a - 3\vec b)$, externally in the ratio $1:2$ . Also, show that $P$ is the mid point of the line segment $RQ$.
SA Q10 The two adjacent sides of a parallelogram are $2\hat i - 4\hat j + 5\hat k$ and $\hat i - 2\hat j - 3\hat k$. Find the unit vector parallel to its diagonal. Also, find its area. SA Q11 Show that the direction cosines of a vector equally inclined to the axes $OX,OY$ and $OZ$ are$\cfrac{1}{{\sqrt 3 }},\cfrac{1}{{\sqrt 3 }},\cfrac{1}{{\sqrt 3 }}$.
SA Q12 Let $\overrightarrow a = \hat i + 4\hat j + 2\hat k,\overrightarrow b = 3\hat i - 2\hat j + 7\hat k$ and $\vec c = 2\hat i - \hat j + 4\hat k$. Find a vector $\vec d$ which is perpendicular to both $\vec a$ and $\bar b$, and $\vec c \cdot \vec d = 15 \cdot$
SA Q13 The scalar product of the vector $\widehat i + \widehat j + \widehat k$ with a unit vector along the sum of vectors$2\hat i + 4\hat j - 5\hat k$ and $\lambda \hat i + 2\widehat j + 3\hat k$ is equal to one. Find the value of $\lambda$.
SA Q14 If $\vec a,\overrightarrow b ,\overrightarrow c$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec a + \vec b + \vec c$ is equally inclined to $\vec a,\vec b$ and $\vec c$. SA Q15 Prove that $(\vec a + \vec b) \cdot (\vec a + \vec b) = |\vec a{|^2} + |\vec b{|^2}$, if and only if $\vec a,\vec b$ are perpendicular, given $\vec a \ne \vec 0,\vec b \ne \vec 0$. SA Q16 If $\theta$ is the angle between two vectors $\vec a$ and $\vec b$, then $\vec a \cdot \vec b \ge 0$ only

• $0 < \theta < \cfrac{\pi }{2}$

• $0 \le \theta \le \cfrac{\pi }{2}$

• $0 < \theta < \pi$

• $0 \le \theta \le \pi$

SA Q17 Let $\vec a$ and $\vec b$ be two unit vectors and $\theta$ is the angle between them. Then, $\vec a + \vec b$ is a unit vector if

• $\theta = \cfrac{\pi }{4}$

• $\theta = \cfrac{\pi }{3}$

• $\theta = \cfrac{\pi }{2}$

• $\theta = \cfrac{{2\pi }}{3}$

[NCERT,Page 459,Misc,Q.No 17] SA Q18 The value of $\hat i \cdot (\hat j \times \hat k) + \hat j \cdot (\hat i \times \hat k) + \hat k \cdot (\hat i \times \hat j)$ is

• $0$

• $- 1$

• $1$

• $3$

SA Q19 If $\theta$ is the angle between any two vectors $\vec a$ and $\vec b$, then $|\vec a \cdot \vec b| = |\vec a \times \vec b|$ when $\theta$ is equal to

•
• $0$

• $\cfrac{\pi }{4}$

• $\cfrac{\pi }{2}$

• $\pi$

SA