91Ó°ÊÓ

Find a parametrization for the line segment between points \(A\) and \(B\) in each of the following questions. (i) \(A(0,0,0), B(1,1,3)\) (ii) \(A(-1,0,1), B(1,1,-2)\) (iii) \(A(1,0,-1), B(0,3,0)\)

Find \(k\) such that the angle between the vectors \(\left(\begin{array}{l}k \\\ 1 \\ 1\end{array}\right)\) and \(\left(\begin{array}{l}1 \\ k \\\ 1\end{array}\right)\) is \(\frac{\pi}{3}\)

For what value(s) of \(m\) are the following four points on the same plane? \(A(m, 3,-2), B(3,4, m), C(2,0,-2)\) and \(D(4,8,4)\)

Find the acute angle between the given lines or planes. $$3 x+4 y-z=1 \text { and } x-2 y=3$$

$$\text { Find } x \text { and } y \text { such that }\left(\begin{array}{l} 2 \\ x \\ y \end{array}\right) \text { is perpendicular to both }\left(\begin{array}{r} 3 \\ 1 \\ -1 \end{array}\right) \text { and }\left(\begin{array}{r} 4 \\ -1 \\ 2 \end{array}\right)$$

Find the acute angle between the given lines or planes. $$4 x-7 y+z=3 \text { and } 3 x+2 y+2 z=17$$

Find the area of the triangle with the given vertices. $$A(2,6,-1), B(1,1,1), C(3,5,2)$$

Find a vector equation and a set of parametric equations of the line through the point (0,2,3) and parallel to the line \(\mathbf{r}=(\mathbf{i}-2 \mathbf{j})+2 t \mathbf{k}\).

Find a vector equation and a set of parametric equations of the line through the point (1,2,-1) and parallel to the line \(\mathbf{r}=t(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k})\).

$$\text { Find } \alpha \text { such that }|\alpha \mathbf{i}+(\alpha-1) \mathbf{j}+(\alpha+1) \mathbf{k}|=2$$.