List of top Mathematics Questions on Vector basics asked in KEAM

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

If \(O\) is the origin and \(C\) is the midpoint of \(A(-2,1)\) and \(B(4,-3)\), then \(\vec{OC}\) is:

The shortest distance between the lines $\vec{r}=(2\hat{i}+6\hat{j}+3\hat{k})+t(2\hat{i}+3\hat{j}+4\hat{k}), t\in\mathbb{R}$ and $\vec{r}=(2\hat{j}-3\hat{k})+s(\hat{i}+2\hat{j}+3\hat{k}), s\in\mathbb{R}$, is

The straight line $\vec{r}=(2-3t)(\hat{i}+\hat{j})+(6t+1)(\hat{j}+\hat{k})+(12t-11)(\hat{k}+\hat{i}), t\in\mathbb{R}$, is parallel to the vector

The equation of a line passing through (-1,6,5) and (-2,4,3), is

If $\alpha,\beta,\gamma$ are the angles made by the straight line $\frac{x-5}{2}=\frac{y+1}{3}=\frac{z-2}{\sqrt{7}}$ with the x-axis, y-axis and z-axis respectively, then $\cos\alpha, \cos\beta, \cos\gamma$ are respectively,

The position vectors of the vertices of a triangle are $\hat{i}+2\hat{j}-4\hat{k}$, $-\hat{i}-2\hat{j}-4\hat{k}$ and $2\hat{i}+3\hat{j}+5\hat{k}$. The perimeter of the triangle is

Let $|\vec{a}|=2,|\vec{b}|=\sqrt{13+6\sqrt{3}}$ and $|\vec{c}|=3$. If $\vec{a}-\vec{b}+\vec{c}=0$, then the angle between $\vec{a}$ and $\vec{c}$ is

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $|\vec{a}|=2, |\vec{b}|=3, |\vec{c}|=4$ and $\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}=0$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$, then $\vec{a}$ is equal to

The vectors $2\vec{a}+6\vec{b}$ and $3\vec{a}-7\vec{b}$ are position vectors of the points A and B respectively. A point P divides the line segment AB internally in the ratio 3:5. Then $\vec{PB}=$

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?

The direction cosines of the vector $\hat{i}-5\hat{j}+8\hat{k}$ are

The values of $\alpha$ so that $|\alpha\hat{i}+(\alpha+1)\hat{j}+2\hat{k}|=3$, are

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 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

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: