Question

Write the projection of the vector $\stackrel{\rightarrow}{a}\$ = $2\stackrel{\wedge}{i}-\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ on the vector $\stackrel{\rightarrow}{b}\$ = $\stackrel{\wedge}{i}+2\stackrel{\wedge}{j} +2\stackrel{\wedge}{k}$.

Find ‘λ’ when the projection of $\stackrel{\rightarrow}{a}\$ = $λ\stackrel{\wedge}{i}+\stackrel{\wedge}{j} +4\stackrel{\wedge}{k}$ on $\stackrel{\rightarrow}{b}\$ = $2\stackrel{\wedge}{i}+6\stackrel{\wedge}{j} +3\stackrel{\wedge}{k}$ is 4 units.

Find a vector of magnitude 5 units and parallel to the resultant of $\stackrel{\rightarrow}{a}\$ = $2\stackrel{\wedge}{i}+3\stackrel{\wedge}{j} -\stackrel{\wedge}{k}$ and $\stackrel{\rightarrow}{b}\$ = $\stackrel{\wedge}{i}-2\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$.

Write a unit vector in the direction of the sum of vectors : $\stackrel{\rightarrow}{a}\ = 2\stackrel{\wedge}{i}-\stackrel{\wedge}{j} +2\stackrel{\wedge}{k} and \stackrel{\rightarrow}{b}\ = \stackrel{\wedge}{i}+\stackrel{\wedge}{j} +3\stackrel{\wedge}{k}$

Find the projection of vector $\stackrel{\wedge}{i}+3\stackrel{\wedge}{j} +7\stackrel{\wedge}{k}$ on the vector $2\stackrel{\wedge}{i}-3\stackrel{\wedge}{j} +6\stackrel{\wedge}{k}$.

Write the value of λ so that the vectors $\stackrel{\rightarrow}{a}\$ = $2 \stackrel{\wedge}{i}+λ\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ and $\stackrel{\rightarrow}{b}\$ = $\stackrel{\wedge}{i}-2\stackrel{\wedge}{j} +3\stackrel{\wedge}{k}$ are perpendicular to each other ?

Find $\stackrel{\rightarrow}{a}$.$\stackrel{\rightarrow}{b}$,If $\stackrel{\rightarrow}{a}$ = $-\stackrel{\wedge}{i}+\stackrel{\wedge}{j} -2\stackrel{\wedge}{k}$ and $\stackrel{\rightarrow}{b}$ = $2\stackrel{\wedge}{i}+3\stackrel{\wedge}{j} -\stackrel{\wedge}{k}$.