Question

1- Two vectors are given as u = 2 – 5j and v=-{+3j. a- Find the vector 2u +3v (by calculation, not by drawing). (4 pts) b- Find the magnitudes luand il of the two vectors. (4 pts) c- Calculate the scalar product u•v. (5 pts) d- Find the angle between the vectors u and v. (6 pts) - Calculate the vector product uxv. (6 pts)

Answer 1

Given:

$\vec u=2\hat i-5\hat j\\ \vec v=-\hat i+3\hat j$

a)

$2\vec u+3\vec v=2(2\hat i-5\hat j)+3(-\hat i+3\hat j)\\ =4\hat i-10\hat j-3\hat i+9\hat j\\ =\hat i-\hat j$

b)

The magnitudes are:

$|\vec u|=\sqrt{2^2+(-5^2)}=5.4\\ |\vec v|=\sqrt{(-1)^2+3^2}=3.2$

c)

Dot product between the vectors is,

$\vec u.\vec v=(2\hat i-5\hat j).(-\hat i+3\hat j)\\ =-2-15\\ =-17$

d)

Angle between two vectors is,

$\theta=\cos^{-1}(\frac{\vec u.\vec v}{|\vec u||\vec v|})\\ =\cos^{-1}(\frac{-17}{5.4\times3.2})\\ =170^0$

e)

Cross product between the vectors is,

$\vec u\times\vec v=\begin{vmatrix} \hat i &\hat j &\hat k \\ 2 &-5 &0 \\ -1 &3 &0 \end{vmatrix}\\ =\hat i(0)-\hat j(0)+\hat k(6-5)\\ =-\hat k$