91Ó°ÊÓ

Find the intersection of the two planes with equations \(3(x-1)+2 y+(z+1)=0\) andN \((x-1)+4 y-(z+1)=0.\)

Let \(\mathbf{u}=(1,2), \mathbf{v}=(-3,4),\) and \(\mathbf{w}=(5,0)\). (a) Draw these vectors in \(\mathbb{R}^{2}\). (b) Find scalars \(\lambda_{1}\) and \(\lambda_{2}\) such that \(\mathbf{w}=\lambda_{1} \mathbf{u}+\lambda_{2} \mathbf{v}\).

(a) Find all points \(\mathbf{p} \in \mathbb{R}^{3}\) that have the same representation in both Cartesian and spherical coordinates. (b) Find all points \(\mathbf{p} \in \mathbb{R}^{3}\) that have the same representation in both Cartesian and cylindrical coordinates.

What restrictions must be made on the scalar \(b\) so that the vector \(2 \mathbf{i}+b \mathbf{j}\) is orthogonal to \((\mathrm{a})-3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}\) and (b) \(\mathbf{k} ?\)

Assuming the law det \((A B)=(\operatorname{det} A)(\operatorname{det} B),\) verify that (det \(A\) ) (det \(A^{-1}\) ) \(=1\) and conclude that if \(A\) has an inverse, then det \(A \neq 0\).

(a) Prove the two triple-vector-product identities $$ (a \times b) \times c=(a \cdot c) b-(b \cdot c) a $$ and a \(\times(b \times c)=(a \cdot c) b-(a \cdot b) c\) (b) Prove \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w}=\mathbf{u} \times(\mathbf{v} \times \mathbf{w})\) if and only if \((u \times w) \times v=0\) (c) Also prove that \((\mathbf{a} \times \mathbf{v}) \times \mathbf{w}+(\mathbf{v} \times \mathbf{w}) \times \mathbf{u}+(\mathbf{w} \times \mathbf{u}) \times \mathbf{v}=\mathbf{0}\) (called the Jacobi identity).

Vectors \(\mathbf{v}\) and \(\mathbf{w}\) are sides of an equilateral triangle whose sides have length \(1 .\) Compute \(\mathbf{v} \cdot \mathbf{w}\)

Suppose \(A, B,\) and \(C\) are vertices of a triangle. Find \(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}\).

Find the points of intersection of the line \(x=3+2 t, y=7+8 t, z=-2+t,\) that \(\mathrm{is}, \mathrm{I}(t)=\) \((3+2 t, 7+8 t,-2+t),\) with the coordinate planes.

Find two \(2 \times 2\) matrices \(A\) and \(B\) such that \(A B=0\) but \(B A \neq 0\).