91Ó°ÊÓ

Determine whether the lines \(l\), and \(l_{2}\) are parallel, skew or intersecting. If they intersect, find the coordinates of the point of intersection. $$l_{1}: \mathbf{r}=\mathbf{i}+2 \mathbf{j}+\lambda(-6 \mathbf{i}+9 \mathbf{j}-3 \mathbf{k}), l_{2}: \mathbf{r}=2 \mathbf{i}+3 \mathbf{j}+m(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k})$$

Find the line of intersection between the given planes. $$x=10 \text { and } x+y+z=3$$

Find \(m\) such that the following vectors are coplanar; otherwise, show that it is not possible. $$\mathbf{u}=(2,-3,2 m), \mathbf{v}=(m,-3,1), \mathbf{w}=(1,3,-2)$$

Find the line of intersection between the given planes. $$2 x-y+z=5 \text { and } x+y-z=4$$

Determine whether the lines \(l\), and \(l_{2}\) are parallel, skew or intersecting. If they intersect, find the coordinates of the point of intersection. $$\frac{x-2}{5}=y-1=\frac{z-2}{3} \text { and } \frac{x+4}{3}=\frac{7-y}{3}=\frac{10-z}{4}$$

Let \(u=i+m j+k\) and \(v=2 i-j+n k\) Compute all values of \(m\) and \(n\) for which \(\mathbf{u} \perp \mathbf{v}\) and \(|\mathbf{u}|=|\mathbf{v}|\)

Determine whether the lines \(l\), and \(l_{2}\) are parallel, skew or intersecting. If they intersect, find the coordinates of the point of intersection. $$x=1+t, y=2-2 t, z=t+5 \text { and } x=2+2 t, y=5-9 t, z=2+6 t$$

Show that \(\frac{\pi}{4}, \frac{\pi}{6}, \frac{2 \pi}{3}\) cannot be the direction angles of a vector.

Find the point on the line $$ r=2 \mathbf{i}+3 \mathbf{j}+\mathbf{k}+t(-3 \mathbf{i}+\mathbf{j}+\mathbf{k}) $$ that is closest to the origin. (Hint: use the parametric form and the distance formula and minimize the distance using derivatives!)

Find the point on the line $$ r=4 j+5 k+t(1-3 j+k) $$ that is closest to the origin.