Question

of meters. 30. Given these two vectors A (3.00)i + (7.00)j - (4. 00) k and B (2.00)i - (4.50)j + (3. 00) k: a. Calculate A. B Calculate A >< B b. Calculate the angle between A and B Determine by calculation if the vector c. is perpendicular to B. d. Calculate a unit vector that points in the direction of the vector-A

Answer 1

a)

$\vec A \cdot \vec B=(3\times 2)-(7\times 4.5)-(4\times 12)=6-31.5&plus;12=-37.5$

$\vec A\times \vec B=\begin{vmatrix} \hat i & \hat j & \hat k\\ 3& 7 & -4\\ 2& -4.5 & 3 \end{vmatrix}=3\hat i-17\hat j-27.5\hat k$

b) Angle between A and B is

$\theta=\cos^{-1}\frac{\vec A \cdot \vec B}{|\vec A||\vec B|}=\cos^{-1}\frac{-37.5}{\sqrt{3^2&plus;7^2&plus;4^2}\sqrt{2^2&plus;4.5^2&plus;3^2}}=\cos^{-1}(-0.756)$

Hence

$\theta=139.11^o$

c) If dot product of B and C is zero, then they are perpendicular

$\vec B\cdot \vec C=(2\times 7.8)-(4.5\times 4.6)-(3\times 1.7)=-10.2$

Therefore C is not perpendicular B

d) Unit vector in the direction of -A is

$-\hat A=-\frac{\vec A}{|\vec A|}=-\frac{3\hat i&plus;7\hat j-4\hat k}{\sqrt{74}}=-0.349\hat i-0.814\hat j&plus;0.465\hat k$