Find a vector $\stackrel{\rightarrow}{a}\$ of magnitude ${\mathfrak{5}}{\sqrt{2}}$ making an angle of $\frac{\pi}{4}$ with x-axis, $\frac{\pi}{2}$ with y-axis and an acute angle θ with z-axis.

If a unit vector $\stackrel{\rightarrow}{a}\$ makes an angle $\frac{\pi}{3}$ with $\stackrel{\wedge}{i},$ $\frac{\pi}{4}$ with $\stackrel{\wedge}{j}$ and an acute angle θ with $\stackrel{\wedge}{k}$, then find the value of θ.

P and Q are two points with position vectors $3\stackrel{\rightarrow}{a}-2\stackrel{\rightarrow}{b}$ and $\stackrel{\rightarrow}{a}+\stackrel{\rightarrow}{b}$ respectively. Write the position vector of a point R which divides the line segment PQ externally in the ratio 2 : 1.

L and M are two points with position vectors $2\stackrel{\rightarrow}{a}-\stackrel{\rightarrow}{b}$ and $\stackrel{\rightarrow}{a}+2\stackrel{\rightarrow}{b}$ respectively. Write the position vectors of a point N which divides the line segment LM in the ratio 2:1 externally.

Find the scalar components of the vector $\stackrel{\rightarrow}{A B}$ with initial point A(2,1) and terminal point B (– 5, 7).

If a line has direction ratios 2, – 1, – 2, then what are its direction cosines ?

If $\stackrel{\rightarrow}{a}\$ and $\stackrel{\rightarrow}{b}\$ denote the position vectors of points A and B respectively and C is a point on AB such that AC = 2 CB, then write the position vector of C.

If $\stackrel{\rightarrow}{a}\$ =$4\stackrel{\wedge}{i}-\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ and $\stackrel{\rightarrow}{b}\$ =$2\stackrel{\wedge}{i}-2\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ then find a unit vector parallel to the vector $\stackrel{\rightarrow}{a}+\stackrel{\rightarrow}{b}$

Give an example of vectors $\stackrel{\rightarrow}{a}\$ and $\stackrel{\rightarrow}{b}\$ such that $\vert{\vec{b}}\vert=$ $\vert{\vec{b}}\vert$ but $\stackrel{\rightarrow}{a}\neq \stackrel{\rightarrow}{b}\;.$  

Write a unit vector in the direction of vector $\stackrel{\rightarrow}{PQ}\$ where $\stackrel{\rightarrow}{p}\$ and $\stackrel{\rightarrow}{q}\$ are the points (1, 3, 0) and (4, 5, 6),respectively,

For what values of $\stackrel{\rightarrow}{a}\$, the vectors $2\stackrel{\wedge}{i}-3\stackrel{\wedge}{j} +4\stackrel{\wedge}{k}$ and $a\,{\hat{\boldsymbol{i}}}+6\,{\hat{\boldsymbol{j}}}-8\,{\hat{\boldsymbol{k}}}$ are collinear?

If A, B and C are the vertices of a $\Delta A B C$, then what is the value of ${\vec{AB}}+{\vec{BC}}+{\vec{C A}}$ ?

Find the area of the parallelogram whose diagonals are represented by the vectors $\stackrel{\rightarrow}{a}\$ = $2{\hat{i}}-3{\hat{j}}+4{\hat{k}}$ and ${\vec{b}}=2{\hat{i}}-{\hat{j}}+2{\hat{k}}$

Find the angle between the vectors ${\vec{a}}={\hat{i}}+{\hat{j}}-{\hat{k}}$ and ${\vec{b}}={\hat{i}}-{\hat{j}}+{\hat{k}}$

if $|\stackrel{\rightarrow}{a}+\stackrel{\rightarrow}{b}|$ =60, $|\stackrel{\rightarrow}{a}-\stackrel{\rightarrow}{b}|$=40 and $\stackrel{\rightarrow}{a}\$=22,then find $\stackrel{\rightarrow}{b}\$.