Vector Algebra Set-6

Q1. Let $\mathbf{a}=(2,-1,1)$ and $\mathbf{b}=(1,2,2)$. Find the real scalar $k$ for which $|\mathbf{a}+k\mathbf{b}|$ is minimum.

Q2. The lines $L_1:\ \mathbf{r}=(1,2,3)+t(2,-1,1)$ and $L_2:\ \mathbf{r}=(3,0,1)+s(1,1,k)$ intersect for some real $k$. The value of $k$ for which they intersect is

Q3. Let $\mathbf{a}=(2,1,-1)$ and $\mathbf{b}=(1,0,1)$. A vector $\mathbf{c}$ satisfies $\mathbf{c}\cdot(\mathbf{a}-\mathbf{b})=0$ and the volume of the parallelepiped formed by $\mathbf{a},\mathbf{b},\mathbf{c}$ is $6$, i.e. $|(\mathbf{a}\times\mathbf{b})\cdot\mathbf{c}|=6$. Among all such $\mathbf{c}$, the minimum possible value of $|\mathbf{c}|$ is

Q4. Find the shortest distance between the skew lines $L_1:\ \mathbf{r}=(1,2,-1)+t(1,-1,2)$ and $L_2:\ \mathbf{r}=(2,0,1)+s(2,1,-1)$.

Q5. Let non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy $|\mathbf{a}|=3,\ |\mathbf{b}|=4$ and $|\mathbf{a}-2\mathbf{b}|=7$. The angle between $\mathbf{a}$ and $\mathbf{b}$ is

Q6. Let $\vec{a}=2\hat{i}-\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+4\hat{j}-2\hat{k}$. The scalar projection of $\vec{a}$ on $\vec{b}$ is

Q7. The shortest distance between the skew lines $L_1:\ \vec r=(1,0,2)+\lambda(1,2,-1)$ and $L_2:\ \vec r=(2,-1,3)+\mu(2,1,1)$ is

Q8. For which value of $k$ are the vectors $\vec{u}=(1,k,2)$, $\vec{v}=(2,1,k)$ and $\vec{w}=(k,2,1)$ pairwise orthogonal?

Q9. Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}-\hat{j}+4\hat{k}$. The plane $\Pi$ passes through the origin and contains the vectors $\vec{a}$ and $\vec{b}$. The perpendicular distance from the point $P(3,1,2)$ to the plane $\Pi$ is

Q10. The plane $\Pi$ passes through points $A(1,0,1)$, $B(2,1,3)$ and $C(0,2,1)$. The foot of the perpendicular from $P(3,1,2)$ to the plane $\Pi$ is

Q11. If $\mathbf{a}=3\mathbf{i}-\mathbf{j}+2\mathbf{k}$ and $\mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k}$, a unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ with positive $k$-component is:

Q12. Let $\mathbf{a}=2\mathbf{i}+\mathbf{j}-\mathbf{k}$ and $\mathbf{b}=\mathbf{i}-\mathbf{j}+2\mathbf{k}$. For what value(s) of $\lambda$ is $\mathbf{x}=\mathbf{a}+\lambda\mathbf{b}$ coplanar with $\mathbf{p}=\mathbf{i}+2\mathbf{j}-\mathbf{k}$ and $\mathbf{q}=3\mathbf{i}-\mathbf{j}+4\mathbf{k}$?

Q13. Given $\mathbf{a}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}$, $\mathbf{b}=2\mathbf{i}-\mathbf{j}+\mathbf{k}$ and $\mathbf{c}=\mathbf{i}-\mathbf{j}+4\mathbf{k}$, find a vector $\mathbf{x}=p\mathbf{a}+q\mathbf{b}$ such that $\mathbf{x}\cdot\mathbf{c}=0$, $|\mathbf{x}|=3$, and the first component of $\mathbf{x}$ is positive.

Q14. Lines $L_1: \mathbf{r}=(1,0,2)+s(2,1,-1)$ and $L_2: \mathbf{r}=(3,4,1)+t(1,-1,2)$ are skew. Find the points $P$ on $L_1$ and $Q$ on $L_2$ such that $PQ$ is the shortest segment joining the lines, and compute the shortest distance $PQ$.

Q15. Let non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ be non-parallel. Consider the statements:
P: "There does not exist a vector $\mathbf{w}$ such that

"
R: "If $\mathbf{w}\cdot\mathbf{b}=0$ then $\mathbf{w}\times(\mathbf{a}\times\mathbf{b})$ is parallel to $\mathbf{b}$, so it cannot equal $\mathbf{a}$ since $\mathbf{a}$ is not parallel to $\mathbf{b}$."
Which of the following is correct?