If , and are three vectors such that + + = 0 and | | = 2 , | | = 3 and | | = 5 , then the value of + + is

Here, + + = and 2 = 4, 2 = 9, 2 = 25 ( + + ) ( + + ) = + a b + a c + b a + b 2 + b c + c a + c b + c c 2 = 0 4 + 9 + 25 + 2( + + ) = 0 a b + b c + c a = 2 = - 19

Vector Algebra — Class 12 Maths Solution

exemplar objective MCQ NCERT,Exemp,Q.No.31,Page.218

Question

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

(a) 0
(b) 1
(c) -19 ✓ Correct
(d) 38

Step-by-step Solution

Here, $\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}} + \overrightarrow {\rm{c}} = \overrightarrow {\rm{0}}$

and ${{\rm{a}}^{\vec 2}} = 4,\overrightarrow {{{\rm{b}}^2}} = 9,\overrightarrow {{{\rm{c}}^2}} = 25$

$\therefore (\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}} + \overrightarrow {\rm{c}} ) \cdot (\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}} + \overrightarrow {\rm{c}} ) = \overrightarrow {\rm{0}}$

$\Rightarrow$ $\overrightarrow {{a^2}} + \vec a \cdot \vec b + \vec a \cdot \vec c + \vec b \cdot \vec a + {\vec b^2} + \vec b \cdot \vec c + \vec c \cdot \vec a + \vec c \cdot \vec b + c{\vec c^2} = \vec 0$

$\Rightarrow$
$\Rightarrow$ $4 + 9 + 25 + 2(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} + \overrightarrow {\rm{b}} \cdot \overrightarrow {\rm{c}} + \overrightarrow {\rm{c}} \cdot \overrightarrow {\rm{a}} ) = 0$

$\Rightarrow$ $\vec a \cdot \vec b + \vec b \cdot \vec c + \vec c \cdot \vec a = \frac{{ - 38}}{2} = - 19$

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NCERT & Exemplar solution for CBSE Class 12 Mathematics, Vector Algebra. Curated by Sachin Sharma. Free for all students.