List of top Mathematics Questions on Vectors

Let $ \mathbf{a} = e\hat{i} + 3\hat{j} + 4\hat{k}, \mathbf{b} = \hat{i} + 2\hat{j} - \hat{k}, \mathbf{c} = 3\hat{i} + \hat{j} + \lambda \hat{k} $ be the co-terminal edges of a parallelepiped whose volume is 5 units. Then the value of $ \lambda $ is:

Let $\vec{a}$ and $\vec{c}$ be unit vectors such that the angle between them is $\cos^{-1} \left( \frac{1}{4} \right)$. If $\vec{b} = 2\vec{c} + \lambda \vec{a}$. Where $\lambda > 0$ and $|\vec{b}| = 4$, then $\lambda$ is equal to:

If $ \mathbf{a} = \hat{i} + \hat{j} + \hat{k}, \, \mathbf{b} = 2\hat{i} - \hat{j} + 3\hat{k}, \, \mathbf{c} = \hat{i} - 2\hat{j} + \hat{k} $, $\text{ then a vector of magnitude }$ $ \sqrt{22} $ $\text{ which is parallel to }$ $ 2\mathbf{a} - \mathbf{b} + \mathbf{c} $ is:

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}| = 3$, $|\vec{b}| = 4$ and $|\vec{a} + \vec{b}| = 1$, then the value of $|\vec{a} \times \vec{b}|$ is:

If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a} \times \vec{b} = \vec{c}$, $\vec{a} \cdot \vec{c} = 2$ and $\vec{b} \cdot \vec{c} = 1$. If $|\vec{b}| = 1$, then the value of $|\vec{a}|$ is:

The length of the projection of $ \mathbf{a} = 2\hat{i} + 3\hat{j} + \hat{k} $ $\text{ on }$ $ \mathbf{b} = -2\hat{i} + \hat{j} + 2\hat{k} $ $\text{ is equal to:}$

If $ \mathbf{a}, \mathbf{b}, \mathbf{c} $ are the position vectors of the points $ A, B, C $ respectively and \[ 5\mathbf{a} - 3\mathbf{b} - 2\mathbf{c} = \mathbf{0}, \] then find the ratio in which the point $ C $ divides the line segment $ BA $ externally.

The cross product of vector $ \vec{A} $ and vector $ \vec{B} $ has a magnitude of 50 units, where vector $ \vec{A} $ has a magnitude of 10. The angle between vector $ \vec{A} $ and $ \vec{B} $ is 60 degrees. What is the magnitude of vector $ \mathbf{B} $?

Prove by vector method, the angle subtended on a semicircle is a right angle.

Find the area of the parallelogram whose adjacent sides are represented by the vectors $ \vec{a} = 3\hat{i} + \hat{j} + 2\hat{k} $ and $ \vec{b} = \hat{i} + 2\hat{j} - 2\hat{k} $.

Show that the points $ A(2, 3, -4), B(1, -2, 3) $ and $ C(3, 8, -11) $ are collinear.

Let \[ \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}, \quad \vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}, \quad \vec{c} = c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k}, \] show that \[ \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}. \]

Let $ \mathbf{A} = 2\hat{i} + \hat{j} - 2\hat{k} $ and $ \mathbf{B} = \hat{i} + \hat{j} $. If $ \mathbf{C} $ is a vector such that $ |\mathbf{C} - \mathbf{A}| = 3 $ and the angle between $ \mathbf{A} \times \mathbf{B} $ and $ \mathbf{C} $ is $ 30^\circ $, then $ [(\mathbf{A} \times \mathbf{B}) \times \mathbf{C}] = 3 $, the value of $ \mathbf{A} \cdot \mathbf{C} $ is equal to:

If $ \mathbf{a} $ and $ \mathbf{b} $ are unit vectors such that $ 2\mathbf{a} + \mathbf{b} = 3 $, then which of the following statement is true?

If $ \mathbf{a} $ and $ \mathbf{b} $ are vectors in space, given by $ \mathbf{a} = \frac{\hat{i} - 2\hat{j}}{\sqrt{5}} $ and $ \mathbf{b} = \frac{2\hat{i} + \hat{j} + 3\hat{k}}{\sqrt{14}} $, then the value of \[ (2\mathbf{a} + \mathbf{b}) \cdot \left[ (\mathbf{a} \times \mathbf{b}) \times ( \mathbf{a} - 2\mathbf{b} ) \right] \] is:

If a vector having magnitude of 5 units, makes equal angle with each of the three mutually perpendicular axes, then the sum of the magnitude of the projections on each of the axis is

If \[ \mathbf{a} = \hat{i} - \hat{k}, \mathbf{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k}, \mathbf{c} = y\hat{i} + x\hat{j} + (1 + x - y)\hat{k}, \] $\text{then }$ [$\mathbf{a}$ $\mathbf{b}$ $\mathbf{c}$] $\text{ depends on:}$

If $ \theta = \cos^{-1} \left( \frac{3}{\sqrt{20}} \right) $ is the angle between $ \mathbf{a} = \hat{i} - 2x\hat{j} + 2y\hat{k} $ and $ \mathbf{b} = x\hat{i} + \hat{j} + y\hat{k} $, then the possible values at $ (x, y) $ that lie on the locus are:

Let a, b, c be distinct non-negative numbers. If the vectors $ a\hat{i} + a\hat{j} + c\hat{k} $, $ \hat{i} + \hat{k} $ and $ c\hat{i} + c\hat{j} + b\hat{k} $ lie in a plane, then c is

If $ (\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c}) $, then

If $ \vec{a} = \lambda\hat{i} + \hat{j} - 2\hat{k} $, $ \vec{b} = \hat{i} + \lambda\hat{j} - 2\hat{k} $, $ \vec{c} = \hat{i} + \hat{j} + \hat{k} $ and $ [\vec{a}\vec{b}\vec{c}] = 7 $, then the values of $ \lambda $ are

Let $ \vec{a} = 2\hat{i} + 2\hat{j} + \hat{k} $ and $ \vec{b} $ be another vector such that $ \vec{a} \cdot \vec{b} = 14 $ and $ \vec{a} \times \vec{b} = 3\hat{i} + \hat{j} - 8\hat{k} $. The vector $ \vec{b} $ is

If the volume of the parallelepiped whose adjacent edges are $ \vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k} $, $ \vec{b} = \hat{i} + \alpha\hat{j} + 2\hat{k} $, $ \vec{c} = \hat{i} + 2\hat{j} + \alpha\hat{k} $ is 15, then $ \alpha $ is equal to