Vectors Questions

Find a vector in the direction of vector $\vec{a}=3\hat{i}-4\hat{j}$ that has a magnitude of $10$ units.

The vector component of $\vec{P}=2{\hat{a}}_x-{\hat{a}}_z$ along $\vec{Q}=2{\hat{a}}_x-{\hat{a}}_y+2{\hat{a}}_z$ is:

The equation of a plane passing through the point with position vector $\vec{a}$ and perpendicular to vector $\vec{b}$ is:

Find the area of the parallelogram whose adjacent sides are given by the vectors $(3,1,4)$ and $(1,-1,1)$.

What is the perpendicular distance from the point $(2,3,4)$ to the line $\frac{x-0}{1}=\frac{y-0}{0}=\frac{z-0}{0}$?

If the angle between $\vec{a}$ and $\vec{b}$ is $\frac{2\pi }{3}$ and the projection of $\vec{a}$ in the direction of $\vec{b}$ is $-2$, then $|\vec{a}|$ is:

A unit vector is coplanar to the vectors $\vec{a}=\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$ and perpendicular to the vector $\vec{c}=\hat{i}+\hat{j}+\hat{k}$. It is:

For all real $x$, the vectors $\vec{a}=cx\hat{i}-6\hat{j}+3\hat{k}$ and $\vec{b}=x\hat{i}+2\hat{j}+2cx\hat{k}$ make an obtuse angle with each other. Then the value of $c$ must satisfy:

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

Suppose $\vec{v}=2\hat{i}+\hat{j}-\hat{k}$ and $\vec{w}=\hat{i}+3\hat{k}$. If $\vec{u}$ is a unit vector, then the maximum value of the scalar triple product $[\vec{u}\vec{v}\vec{w}]$ is:

A straight line with direction cosines $\{0,0,1\}$ is:

The magnitude of the projection of the vector $2\hat{i}+3\hat{j}+\hat{k}$ on the vector perpendicular to the plane containing the vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$ is:

The vector $\vec{a}\times (\vec{b}\times \vec{a})$ is coplanar with:

The vectors $\overrightarrow{AB}=3\hat{i}+4\hat{k}$ and $\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle $ABC$. The length of the median through $A$ is:

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:

The vectors $\vec{a}=x\hat{i}+y\hat{j}+z\hat{k}$ and $\vec{b}=\hat{j}$ are such that $\vec{a},\vec{c},\vec{b}$ form a right-handed system. Then $\vec{c}$ is:

If $\vec{a}=4\hat{i}+6\hat{j}$ and $\vec{b}=3\hat{j}+4\hat{k}$ then the component of $\vec{a}$ along $\vec{b}$ is:

If $\vec{a}=2\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\vec{c}=3\hat{i}+\hat{j}$ are such that $\vec{a}+\gamma \vec{b}$ is perpendicular to $\vec{c}$, then determine the value of $\gamma$?

A unit vector $\vec{a}$ makes angle $\pi /3$ with $\hat{i}$, $\pi /4$ with $\hat{j}$ and an acute angle $\theta$ with $\hat{k}$.I. The value of $\theta$ is $\pi /3$.II. The components of $\vec{a}$ are $(1/2,1/2,1/2)$.Which of the above statement(s) is/are correct?

If the vertices $A,B,C$ of a triangle $ABC$ are $(1,1,3)$, $(-1,0,0)$, $(0,1,2)$ respectively, then determine $\angle ABC$.

In a triangle $ABC$, $|\overrightarrow{BC}|=8$, $|\overrightarrow{CA}|=7$, $|\overrightarrow{AB}|=10$, then the projection of the $\overrightarrow{AB}$ on $\overrightarrow{AC}$ is equal to:

The unit vector perpendicular to line $L$ (DRs: $1,-3,2$) and the normal to the plane $P_1:x+y+z-6=0$ is:

Which of the following is true?

Determine $|\vec{a}|$ and $|\vec{b}|$, if $(\vec{a}+\vec{b})\cdot (\vec{a}-\vec{b})=8$ and $|\vec{a}|=8|\vec{b}|$?

The number of vectors of unit length perpendicular to vectors $\vec{a}=(1,1,0)$ and $\vec{b}=(0,1,1)$ is:

If $P=\hat{i}+3\hat{j}-2\hat{k}$ and $Q=3\hat{i}+\hat{j}-2\hat{k}$, then what is the position vector of the bisector of $\angle POQ$?

Given $\vec{a}=2\hat{i}$, $\vec{b}=\hat{i}+\hat{j}+\hat{k}$, $\vec{c}=2\hat{k}$ and $\vec{x}-\vec{w}=\vec{v}$ where $\vec{w}=\vec{b}-\vec{a}$ and $\vec{v}=\vec{b}-\vec{c}$. Then $\vec{x}$ is:

The projection of a vector on another vector is: