91Ó°ÊÓ

Find \((a) u \cdot v\) and \((b)\) the angle between \(u\) and \(v\) to the nearest degree. $$\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}$$

If the vector \(v\) has initial point \(P,\) what is its terminal point? $$\mathbf{v}=\langle 23,-5,12\rangle, P(-6,4,2)$$

Two vectors a and b are given. (a) Find a vector perpendicular to both a and b. (b) Find a unit vector perpendicular to both a and b. $$\mathbf{a}=\langle 2,5,3\rangle, \quad \mathbf{b}=\langle 3,-2,-1\rangle$$

Find \((a) u \cdot v\) and \((b)\) the angle between \(u\) and \(v\) to the nearest degree. $$\mathbf{u}=-5 \mathbf{j}, \quad \mathbf{v}=-\mathbf{i}-\sqrt{3} \mathbf{j}$$

Find parametric equations for the line that passes through the points \(P\) and \(Q\) $$P(1,1,0), \quad Q(0,2,2)$$

Two vectors a and b are given. (a) Find a vector perpendicular to both a and b. (b) Find a unit vector perpendicular to both a and b. $$\mathbf{a}=\frac{1}{2} \mathbf{i}-\mathbf{j}+\frac{2}{3} \mathbf{k}, \quad \mathbf{b}=6 \mathbf{i}-12 \mathbf{j}-6 \mathbf{k}$$

Find the magnitude of the given vector. $$\langle- 2,1,2\rangle$$

Find an equation of a sphere with the given radius \(r\) and center \(C\). $$r=5 ; \quad C(2,-5,3)$$

Find the magnitude of the given vector. $$\langle 5,0,-12\rangle$$

Find parametric equations for the line that passes through the points \(P\) and \(Q\) $$P(3,3,3), \quad Q(7,0,0)$$