List of top Mathematics Questions on Vectors

If vector $ \mathbf{a} = 3 \hat{i} + 2 \hat{j} - \hat{k} $ \text{ and } $ \mathbf{b} = \hat{i} - \hat{j} + \hat{k} $, then which of the following is correct?

The unit vector perpendicular to the vectors $ \hat{i} - \hat{j} $ and $ \hat{i} + \hat{j} $ is:

If $ \vec{\alpha} = \hat{i} - 4 \hat{j} + 9 \hat{k} $ and $ \vec{\beta} = 2\hat{i} - \hat{j} + \lambda \hat{k} $ are two mutually parallel vectors, then $ \lambda $ is equal to:

During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by $ \vec{B} = 2\hat{i} + \hat{j} $, $ \vec{W} = 6\hat{i} + 12\hat{j} $ and $ \vec{F} = 12\hat{i} + 1\hat{j} $ respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.

The diagonals of a parallelogram are given by $ \mathbf{a} = 2 \hat{i} - \hat{j} + \hat{k} $ and $ \mathbf{b} = \hat{i} + 3 \hat{j} - \hat{k}$ . Find the area of the parallelogram.

Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors $ \mathbf{a} = 3 \hat{i} + \hat{j} + 2 \hat{k} $ and $ \mathbf{b} = 2 \hat{i} - 2 \hat{j} + 4 \hat{k} $. Determine the angle formed between the kite strings. Assume there is no slack in the strings.

Verify that lines given by $ \vec{r} = (1 - \lambda) \hat{i + (\lambda - 2) \hat{j} + (3 - 2\lambda) \hat{k} $ and $ \vec{r} = (\mu + 1) \hat{i} + (2\mu - 1) \hat{j} - (2\mu + 1) \hat{k} $ are skew lines. Hence, find shortest distance between the lines.}

If $ \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 $, $ |\overrightarrow{a}| = \sqrt{37} $, $ |\overrightarrow{b}| = 3 $, and $ |\overrightarrow{c}| = 4 $, then the angle between $ \overrightarrow{b} $ and $ \overrightarrow{c} $ is:

If the projection of $ \mathbf{a} = \alpha \hat{i} + \hat{j} + 4 \hat{k} $ on $ \mathbf{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k} $ is 4 units, then $ \alpha $ is:

If $ \mathbf{p} $ and $ \mathbf{q} $ are unit vectors, then which of the following values of $ \mathbf{p} \cdot \mathbf{q} $ is not possible?

If a line makes angles of $ \frac{3\pi}{4} $ and $ \frac{\pi}{3} $ with the positive directions of $ x $, $ y $, and $ z $-axes respectively, then $ \theta $ is:

Let $\mathbf{a} = 4\mathbf{i} + 3\mathbf{j}$ and $\mathbf{b}$ be two perpendicular vectors in the XOY-plane. A vector $\mathbf{c}$ in the same plane and having projections 1 and 2 respectively on $\mathbf{a}$ and $\mathbf{b}$ is

If the vectors $2\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}$, $-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$, and $p\mathbf{i} - 2\mathbf{j} + \mathbf{k}$ are coplanar, then the unit vector in the direction of the vector $9p\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}$ is

If the position vectors of the vertices A, B, C of a triangle are $3\mathbf{i} + 4\mathbf{j} - \mathbf{k}$, $\mathbf{i} + 3\mathbf{j} + \mathbf{k}$, and $5(\mathbf{i} + \mathbf{j} + \mathbf{k})$ respectively, then the magnitude of the altitude drawn from A onto the side BC is

In a right-angled triangle, if the position vector of the vertex having the right angle is $-3\mathbf{i} + 5\mathbf{j} + 2\mathbf{k}$ and the position vector of the midpoint of its hypotenuse is $6\mathbf{i} + 2\mathbf{j} + 5\mathbf{k}$, then the position vector of its centroid is

Assertion (A): For the lines $\mathbf{r} = \mathbf{a} + t \mathbf{b}$ and $\mathbf{r} = \mathbf{p} + s \mathbf{q}$, if $(\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q}) \neq 0$, then the two lines are coplanar. Reason (R): $|(\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q})|$ is $|\mathbf{b} \times \mathbf{q}|$ times the shortest distance between the lines $\mathbf{r} = \mathbf{a} + t \mathbf{b}$ and $\mathbf{r} = \mathbf{p} + s \mathbf{q}$.

If $ 3\overline{i} + \overline{j} + \overline{k} $, $ 2\overline{i} + \overline{k} $, and $ \overline{i} + 5\overline{j} $ are the position vectors of three non-collinear points A, B, C respectively. If the perpendicular drawn from C onto $ \overline{AB} $ meets $ \overline{AB} $ at the point $ a\overline{i} + b\overline{j} + c\overline{k} $, then $ a + b + c = $

If $ \overline{a} = \overline{i} - 2\overline{j} - 3\overline{k} $, $ \overline{b} = -2\overline{i} + 3\overline{j} + 4\overline{k} $, $ \overline{c} = 5\overline{i} - 4\overline{j} + 3\overline{k} $, and $ \overline{d} = 3\overline{i} + \overline{j} + 5\overline{k} $ are four vectors, then $ (\overline{a} \times \overline{b}) \cdot (\overline{c} \times \overline{d}) = $

If $ \overline{a} $ is a unit vector, then $ |\overline{a} \times \overline{i}|^2 + |\overline{a} \times \overline{j}|^2 + |\overline{a} \times \overline{k}|^2 = $

If $ \overline{a} = 2\overline{i} - 3\overline{j} + 5\overline{k} $ and $ \overline{b} = -\overline{i} + 3\overline{j} + 3\overline{k} $ are two vectors, then the vector of magnitude 28 units in the direction of the vector $ \overline{a} - \overline{b} $ is:

If $ \vec{a}, \vec{b}, \vec{c} $ are three unit vectors such that $ \vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2} \vec{b} + \frac{1}{2} \vec{c} $, and $ \alpha, \beta $ are the angles between $ \vec{a}, \vec{c} $ and $ \vec{a}, \vec{b} $ respectively, then $ \alpha + \beta = ? $

Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.

A vector $ \mathbf{a} $ makes equal angles with all the three axes. If the magnitude of the vector is $ 5\sqrt{3} $ units, then find $ \mathbf{a} $.

If $ \vec{a} = \hat{i} + 2\hat{j} - \hat{k} $, $ \vec{b} = 2\hat{i} - \hat{j} + 3\hat{k} $, and $ \vec{c} = -\hat{i} + 3\hat{j} + 2\hat{k} $, find $ \vec{a} \cdot (\vec{b} \times \vec{c}) $.

If $ \vec{p} = 3\hat{i} - \hat{j} + 2\hat{k} $, $ \vec{q} = \hat{i} + 4\hat{j} - \hat{k} $, and $ \vec{r} = 2\hat{i} - 3\hat{j} + 5\hat{k} $, find $ \vec{p} \cdot (\vec{q} \times \vec{r}) $.