Vector Algebra Set-1

Q1. Find the scalar projection (component) of the vector $\vec{a}=(2,3,-1)$ on the vector $\vec{b}=(1,1,2)$.

Q2. Vectors $\vec{a}=(1,1,2),\ \vec{b}=(2,-1,1)$ and $\vec{c}=(3,m,4)$ are coplanar. The value of $m$ is

Q3. Find the foot of the perpendicular from $P(3,-1,2)$ to the plane determined by the points $A(1,0,0),\ B(0,1,0),\ C(0,0,1)$.

Q4. Let non-zero vectors $\vec{a},\vec{b}$ satisfy $\vec{a}\times(\vec{b}\times\vec{a})=\vec{b}$. Which of the following must be true?

Q5. The shortest distance between the skew lines $L_1:\ \vec{r}=(0,0,0)+\lambda(1,0,1)$ and $L_2:\ \vec{r}=(1,1,0)+\mu(0,1,1)$ is

Q6. If two vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy $|\mathbf{a}|=3$, $|\mathbf{b}|=4$ and the angle between them is $60^\circ$, then $|3\mathbf{a}-2\mathbf{b}|$ equals

Q7. Let the position vectors of points $A,B,C$ be $\mathbf{a}=(1,0,2)$, $\mathbf{b}=(3,-1,4)$ and $\mathbf{c}=(2,2,1)$. The area of triangle $ABC$ is

Q8. Find the unit vector perpendicular to both $\mathbf{a}=(2,-1,2)$ and $\mathbf{b}=(1,1,0)$ which makes an acute angle with $\mathbf{c}=(1,0,1)$.

Q9. Assertion (A): If non-zero vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ satisfy $\mathbf{a}\times(\mathbf{b}\times\mathbf{c})=\mathbf{0}$, then either $\mathbf{a}$ is orthogonal to both $\mathbf{b}$ and $\mathbf{c}$ or $\mathbf{b}$ and $\mathbf{c}$ are parallel.
Reason (R): Using the vector triple product identity $\mathbf{a}\times(\mathbf{b}\times\mathbf{c})=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{c}(\mathbf{a}\cdot\mathbf{b})$, we get $\mathbf{b}(\mathbf{a}\cdot\mathbf{c})=\mathbf{c}(\mathbf{a}\cdot\mathbf{b})$, which forces the stated alternatives.

Q10. Let non-zero vectors $\mathbf{a},\mathbf{b}$ satisfy $\mathbf{a}\cdot\mathbf{a}=4$, $\mathbf{b}\cdot\mathbf{b}=9$, $\mathbf{a}\cdot\mathbf{b}=2$. Find the vector $\mathbf{x}$ in $\mathrm{span}\{\mathbf{a},\mathbf{b}\}$ such that $\mathbf{x}\cdot\mathbf{a}=6$ and $\mathbf{x}\cdot\mathbf{b}=15$.