Unknowns in basic vectors

Question

Haven't touched vectors for years and need some help with the following

Given $ u = 3i + 2j$ and $ v = 2i + \lambda j$, find values for $\lambda$ such that
a) the angle between $ u$ and $ v$ is $\frac{\pi}{4}$
b) the length of projection of $u$ on $v$ is 2 units.

Any help is appreciated. Thanks!

Answer 1

Hints:

• a) $(u,v)=||u||||v||\cos\theta$
• b) If $w$ is the projection of $u$ on $v$ then $w=\rho v$ for some scalar $\rho$ and $(u-w)\bot v$ (express this by means of inproduct).