Vector Algebra Set-2

Q1. If $\mathbf{a}=\langle 2,-1,3\rangle$ and $\mathbf{b}=\langle 1,2,-2\rangle$, the unit vector in the direction of the projection of $\mathbf{a}$ on $\mathbf{b}$ is:

Q2. The shortest distance between the skew lines $L_1:\ \mathbf{r}=\langle1,0,2\rangle+\lambda\langle1,2,3\rangle$ and $L_2:\ \mathbf{r}=\langle2,-1,1\rangle+\mu\langle2,-1,1\rangle$ is

Q3. Let $A(1,0,0),\ B(0,1,0),\ C(0,0,1)$. A plane passes through $A,B,C$. The foot of the perpendicular from $P(3,2,1)$ to this plane is

Q4. Find the vector $\mathbf{x}$ satisfying both $\mathbf{x}\times\langle1,1,1\rangle=\langle2,-1,-1\rangle$ and $\mathbf{x}\cdot\langle1,1,1\rangle=3$.

Q5. Let $\mathbf{a}=\langle1,1,0\rangle,\ \mathbf{b}=\langle0,1,1\rangle$. Among all vectors $\mathbf{x}$ satisfying $\mathbf{x}\cdot\mathbf{a}=4$ and $\mathbf{x}\cdot\mathbf{b}=5$, the minimum possible value of $|\mathbf{x}|$ is

Q6. The area of the parallelogram determined by the vectors $\mathbf{a}=(2,-1,3)$ and $\mathbf{b}=(1,4,0)$ is

Q7. If $|\mathbf{a}|=3,\;|\mathbf{b}|=4$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$, then the magnitude $|3\mathbf{a}-2\mathbf{b}|$ equals

Q8. Let $\mathbf{a},\mathbf{b}$ be vectors with $\mathbf{a}\cdot\mathbf{a}=4,\;\mathbf{b}\cdot\mathbf{b}=9,\;\mathbf{a}\cdot\mathbf{b}=6$. The scalar $t$ for which $(\mathbf{a}+t\mathbf{b})$ is perpendicular to $(2\mathbf{a}-\mathbf{b})$ is

Q9. Let $\mathbf{a}=(1,2,-1),\;\mathbf{b}=(2,0,3)$ and $\mathbf{c}=(1,1,1)$. A unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ and making an acute angle with $\mathbf{c}$ is

Q10. Points $P,Q,R$ have position vectors $\mathbf{p}=(1,0,2),\;\mathbf{q}=(t,1,1),\;\mathbf{r}=(2,3,t)$. The values of $t$ for which triangle $PQR$ is right-angled at $Q$ are