Mock Test 1 — Vector Algebra

If $\mathbf{a} \cdot \mathbf{b} = 0$ and $\mathbf{a} \times \mathbf{b} = \mathbf{0}$, where $\mathbf{a}$ is a non-zero vector, then $\mathbf{b}$ must be:

If the position vectors of points $A, B, C, D$ are $\hat{i} + \hat{j} + \hat{k}$, $2\hat{i} + 5\hat{j}$, $3\hat{i} + 2\hat{j} - 3\hat{k}$, $\hat{i} - 6\hat{j} - \hat{k}$ respectively, then the angle between $\overrightarrow{AB}$ and $\overrightarrow{CD}$ is:

If $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ are mutually perpendicular unit vectors, then $|\mathbf{a} + \mathbf{b} + \mathbf{c}|$ equals:

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three vectors such that $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$, then $\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}$ equals:

If $\mathbf{a} = \hat{i} + \hat{j} + \hat{k}$ and $\mathbf{b} = \hat{j} - \hat{k}$, then a vector $\mathbf{c}$ satisfying $\mathbf{a} \times \mathbf{c} = \mathbf{b}$ and $\mathbf{a} \cdot \mathbf{c} = 3$ is:

The value of $[\mathbf{a} - \mathbf{b}\,\,\mathbf{b} - \mathbf{c}\,\,\mathbf{c} - \mathbf{a}]$ is:

The vectors $\mathbf{a} = \lambda\hat{i} + \mu\hat{j} + 4\hat{k}$, $\mathbf{b} = 3\hat{i} - 4\hat{j} + 5\hat{k}$ are perpendicular if:

Vectors $\overrightarrow{AB} = 3\hat{i} + \hat{k}$ and $\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of triangle $ABC$. The length of the median through $A$ equals:

If $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ are three non-coplanar vectors and $\mathbf{p}$, $\mathbf{q}$, $\mathbf{r}$ are their reciprocal system, then $(\mathbf{a} + \mathbf{b}) \cdot \mathbf{p} + (\mathbf{b} + \mathbf{c}) \cdot \mathbf{q} + (\mathbf{c} + \mathbf{a}) \cdot \mathbf{r}$ equals:

The shortest distance between the lines $\mathbf{r} = \hat{i} + 2\hat{j} + \hat{k} + \lambda(\hat{i} - \hat{j} + \hat{k})$ and $\mathbf{r} = 2\hat{i} - \hat{j} - \hat{k} + \mu(2\hat{i} + \hat{j} + 2\hat{k})$ is:

If $\mathbf{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\mathbf{b} = \hat{i} - 2\hat{j} + 3\hat{k}$ are perpendicular, find the value of $\lambda$.

The volume of the parallelepiped with adjacent edges $2\hat{i} - 3\hat{j} + 4\hat{k}$, $\hat{i} + 2\hat{j} - \hat{k}$, $3\hat{i} - \hat{j} + 2\hat{k}$ is $V$. Find $V$.

If $\mathbf{a}$ and $\mathbf{b}$ are unit vectors and $\mathbf{a} + 2\mathbf{b}$ and $5\mathbf{a} - 4\mathbf{b}$ are perpendicular, find $60\cos\theta$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.

Assertion (A): The vector triple product satisfies $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$.
Reason (R): The vector triple product is always associative; that is, $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$.

Assertion (A): If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are coplanar non-zero vectors, then $[\mathbf{a}\,\mathbf{b}\,\mathbf{c}] = 0$.
Reason (R): The scalar triple product equals the volume of the parallelepiped formed by the three vectors, and coplanar vectors form a degenerate parallelepiped of zero volume.