List of top Mathematics Questions on Vector basics

Let $\theta$ be the angle between the unit vectors $\hat{a}$ and $\hat{b}$. If $|\hat{a} - \hat{b}| = \frac{\sqrt{3}}{2}$, then the value of $\cos \theta$ is

Let \(|\vec{a}| = 6\) and \(|\vec{b}| = 10\). If \(\vec{a}\) and \(\vec{b}\) make angles \(25^\circ\) and \(85^\circ\), respectively, with the x-axis, then the value of \(|\vec{a} + \vec{b}|\) is equal to

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 $\vec{a} = \vec{i} + \vec{j} + \vec{k}$, $\vec{b} = 4\vec{i} + 3\vec{j} + 4\vec{k}$ and $\vec{c} = \vec{i} + \alpha\vec{j} + \beta\vec{k}$ are linearly dependent vectors and $|\vec{c}| = \sqrt{3}$, then:

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 $\vec{OA}=2\hat{i}+3\hat{j}-5\hat{k}$, $\vec{OB}=3\hat{i}+\hat{j}-2\hat{k}$, $\vec{OC}=6\hat{i}-5\hat{j}+7\hat{k}$ be position vectors of A, B and C. Then ________.

Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors, and \(\theta\) be the angle between them. If \(\vec{a}-\vec{b}\) is a unit vector, then \(\theta\) is equal to

The unit vector that bisects the angle between two vectors \( 2\hat{i}+\hat{j}+2\hat{k} \) and \( \hat{i}+2\hat{j}-2\hat{k} \) is

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

If $|\vec{u}| = 2$ and $\vec{u}$ makes angles of $60^\circ$ and $120^\circ$ with the axes $OX$ and $OY$ at the origin, then $\vec{u} =$

The position vector of the point of intersection of the medians of a triangle, whose vertices are $A(1, 2, 3)$, $B(1, 0, 3)$ and $C(4, 1, -3)$ 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 \( \overrightarrow{QA} + \overrightarrow{QB} + \overrightarrow{QC} = \)