List of top Mathematics Questions on Vector Algebra

The value of expression \(\hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k}\) is

Assertion (A): If $| \mathbf{a} \times \mathbf{b} |^2 + | \mathbf{a} \cdot \mathbf{b} |^2 = 256$ and $| \mathbf{b} | = 8$, then $| \mathbf{a} | = 2$.
Reason (R): $\sin^2 \theta + \cos^2 \theta = 1$ and $| \mathbf{a} \times \mathbf{b} | = | \mathbf{a} | | \mathbf{b} | \sin \theta$ and $ \mathbf{a} \cdot \mathbf{b} = | \mathbf{a} | | \mathbf{b} | \cos \theta$.

Find the angle between the vectors \( -2\hat{i} + \hat{j} + 3\hat{k} \) and \( 3\hat{i} - 2\hat{j} + \hat{k} \).

Find the projection of the vector \(\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}\) on the vector \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\).

Find the unit vector perpendicular to each of the vectors \( (\vec{a} + \vec{b}) \) and \( (\vec{a} - \vec{b}) \) where \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k} \).

The value of expression \( \hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k} \) is

Find the unit vector along the vector \( \vec{a} = 2\hat{i} + 3\hat{j} + \hat{k} \).

Find the value of \( \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) \).

For the two vectors \( \vec{a} \) and \( \vec{b} \), prove that \( |\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}| \) when \( \vec{a} \neq \vec{0} \) and \( \vec{b} \neq \vec{0} \).

Find the area of a triangle whose vertices are A(2, 2, 2), B(2, 1, 3) and C(3, 2, 1).

Find the direction-cosines of the sum of the vectors \( \vec{a} = 3\hat{i} + 4\hat{j} - 3\hat{k} \) and \( \vec{b} = -2\hat{i} - 3\hat{j} + \hat{k} \).

If the position vectors of the points A and B are \(\hat{i}+\hat{j}+\hat{k}\) and \(2\hat{i}+5\hat{j}\) respectively, then find the unit vector along the straight line AB.

Find the area of a triangle \( \triangle ABC \) whose vertices are A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1).

The modulus of two vectors \(\vec{a}\) and \(\vec{b}\) are \(\sqrt{3}\) and 4 respectively, and \(\vec{a} \cdot \vec{b} = 6\). Then find the angle between the vectors \(\vec{a}\) and \(\vec{b}\).

Find the area of a parallelogram whose adjacent sides are the vectors \( \vec{a} = \hat{i} - \hat{j} + 3\hat{k} \) and \( \vec{b} = 2\hat{i} - 7\hat{j} + \hat{k} \).

Obtain the projection of the vector \( \vec{a} = 2\hat{i} + 3\hat{j} + 5\hat{k} \) on the vector \( \vec{b} = \hat{i} + 3\hat{j} + \hat{k} \).

If \( \theta \) be the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \), prove that \( \sin\frac{\theta}{2} = \frac{1}{2} |\hat{a} - \hat{b}| \).

Find the area of the parallelogram whose diagonals are \( \vec{a} = 3\hat{i} + \hat{j} - 2\hat{k} \) and \( \vec{b} = \hat{i} - 3\hat{j} + 4\hat{k} \).

If the coordinates of the points P and Q are respectively (2, 3, 0) and (–1, –2, –4), the vector \( \vec{PQ} \) will be

For two vectors \(\vec{a}\) and \(\vec{b}\) prove that \(|\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|\).

If two vectors \(\vec{a}\) and \(\vec{b}\) are such that \(|\vec{a}| = 2\), \(|\vec{b}| = 3\) and \(\vec{a} \cdot \vec{b} = 4\), find \(|\vec{a} - \vec{b}|\).

Let \(\vec{a} = \hat{i} + 2\hat{j}\) and \(\vec{b} = 2\hat{i} + \hat{j}\). Is \(|\vec{a}| = |\vec{b}|\)? Are the vectors \(\vec{a}\) and \(\vec{b}\) equal?

The direction cosines of the vector \(\hat{i} + \hat{j} - 2\hat{k}\) are

Prove that \(|\vec{a} \cdot \vec{b}| \le |\vec{a}| |\vec{b}|\) is always true for any two vectors \(\vec{a}\) and \(\vec{b}\).