For two vectors $\mathbf{u,v}$ in $\mathbb{R}^n$ euclidean space, given:
- $\|\mathbf{u}\| = 3$
- $\|\mathbf{v}\| = 5$
- $\angle (\mathbf{u,v})=\frac{2\pi}{3}$
Calculate the length of the vectors
- $4\mathbf{u}-\mathbf{v}$
- $2\mathbf{u}-7\mathbf{v}$
I'm not sure how to approach this with the given information
With the formula for the angle between the two vectors being
$$\cos \theta=\frac{\mathbf{u\cdot v}}{\|\mathbf{u}\|\cdot\|\mathbf{v}\|}$$
I already have the denominator, but how do I get the point product of u and v in this case?
A point to start would be most appreciated
Best Answer
$(\mathbf{u}.\mathbf{v})=||\mathbf{u}||||\mathbf{v}||cos(\frac{2\pi}{3})=\frac{-1}{2}||\mathbf{u}||||\mathbf{v}||$
$(4\mathbf{u}-\mathbf{v}).(4\mathbf{u}-\mathbf{v})=16\mathbf{(u.u)-4(\mathbf{u}.\mathbf{v})}-4(\mathbf{v}.\mathbf{u})+(\mathbf{v}.\mathbf{v})=16||\mathbf{u}||^2-8(\mathbf{u}.\mathbf{v})+||\mathbf{v}||^2=16*9-8*\frac{-15}{2}+25=229$ $|(4\mathbf{u}-\mathbf{v})|=\sqrt{229}$