List of top Mathematics Questions on Vector basics

For a scalar function \(\vec{F}(x, y, z) = x^2 + 3y^2 + 2z^2\), the directional derivative at the point P( 1, 2, -1) is the direction of a vector \((\hat{i} + \hat{j} + 2\hat{k})\) is

If $\vec{a} = 2\hat{i} + 2\hat{j} - \hat{k}$, $\vec{b} = \alpha\hat{i} + \beta\hat{j} + 2\hat{k}$ and $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$, then $\alpha + \beta$ is equal to

The value of $\lambda$ for which the vectors $\vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$ are orthogonal is

If $2\hat{i} - \hat{j} + \hat{k} = s(3\hat{i} - 4\hat{j} - 4\hat{k}) + t(\hat{i} - 3\hat{j} - 5\hat{k})$, where $s$ and $t$ are scalars, then $3s + 5t$ is equal to:

Let $\vec{a}, \vec{b}, \vec{c}$ be any three vectors and $m, n$ be scalars. Which one of the following is not true?

Position vector of \( P \) and \( Q \) are \( \hat{i} + 3\hat{j} - 7\hat{k} \) and \( 5\hat{i} - 2\hat{j} + 4\hat{k} \) respectively. Then the cosine of the angle between \( \overrightarrow{PQ} \) and the y-axis is

The magnitude of the projection of the vector \( -\hat{i} + 2\hat{j} - \hat{k} \) on the \(z\)-axis is

If \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} - \hat{j} + 2\hat{k} \), then find the angle \( \theta \) between \( \vec{a} \) and \( \vec{b} \).

Let two non-collinear vectors $\hat{a}$ and $\hat{b}$ form an acute angle. A point P moves, so that at any time t the position vector $\overline{OP}$, where O is origin, is given by $\hat{a}\sin t+\hat{b}\cos t.$ when P is farthest from origin O, let M be the length of OP and $\hat{u}$ be the unit vector along $\overline{OP,}$ then

If $|\vec{a}| = 2$, $|\vec{b}| = 3$, $|\vec{c}| = 5$ and each of the angles between the vectors $\vec{a}$ and $\vec{b}$, $\vec{b}$ and $\vec{c}$, $\vec{c}$ and $\vec{a}$ is $60^\circ$, then the value of $|\vec{a} + \vec{b} + \vec{c}|$ is

The points with position vectors \(60\hat{i}+3\hat{j}, 40\hat{i}-8\hat{j}, a\hat{i}-52\hat{j}\) are collinear if

Suppose $\alpha \hat{i} + \alpha \hat{j} + \gamma \hat{k}$, $\hat{i} + \hat{k}$ and $\gamma \hat{i} + \gamma \hat{j} + \beta \hat{k}$ are coplanar where $\alpha, \beta, \gamma$ are positive constants. Then the product $\alpha\beta$ is

Let \(\mathbf{a} = \mathbf{i} - \mathbf{k}, \quad \mathbf{b} = x\mathbf{i} + \mathbf{j} + (1-x)\mathbf{k}, \quad \mathbf{c} = y\mathbf{i} + x\mathbf{j} + (1+x-y)\mathbf{k}\). Then \([\mathbf{a}, \mathbf{b}, \mathbf{c}]\) depends on:

Let the position vectors of points \(A, B, C\) be \( \vec{a}, \vec{b}, \vec{c} \) respectively. Let \(Q\) be the centroid. Then \( \overrightarrow{QA} + \overrightarrow{QB} + \overrightarrow{QC} = \)

If \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 4\hat{i} + 3\hat{j} + 4\hat{k} \) and \( \vec{c} = \hat{i} + \alpha \hat{j} + \beta \hat{k} \) are coplanar and \( |\vec{c}| = \sqrt{3} \), then

Let \(\vec a,\vec b,\vec c\) be three vectors satisfying \(\vec a\times\vec b=\vec a\times\vec c\), \(|\vec a|=|\vec c|=1\), \(|\vec b|=4\) and \(|\vec b\times\vec c|=\sqrt{15}\). If \(\vec b-2\vec c=\lambda \vec a\), then \(\lambda\) equals

The angle between a normal to the plane \( 2x - y + 2z - 1 = 0 \) and the \( z \)-axis is:

Equation of the plane through the mid-point of the line segment joining the points P(4, 5, -10) and Q(-1, 2, 1) and perpendicular to PQ is:

Let \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \). If \( \vec{b} \) is a vector such that \( \vec{a} \cdot \vec{b} = |\vec{b}|^2 \) and \( |\vec{a} - \vec{b}| = \sqrt{7} \), then \( |\vec{b}| = \)

The angle between a normal to the plane \( 2x - y + 2z - 1 = 0 \) and the \( z \)-axis is:

Equation of the plane through the mid-point of the line segment joining the points P(4, 5, -10) and Q(-1, 2, 1) and perpendicular to PQ is:

Let \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \). If \( \vec{b} \) is a vector such that \( \vec{a} \cdot \vec{b} = |\vec{b}|^2 \) and \( |\vec{a} - \vec{b}| = \sqrt{7} \), then \( |\vec{b}| = \)