(a)

Find the point at which the given lines intersect.

(x, y, z) =

(b)

Find an equation of the plane that contains these lines.

Solve the system of equations

\(\begin{cases}2x+3y=5\\5x−4y=2\end{cases}\)

(a) Find a nonzero vector orthogonal l to the plane the points P, Q, and R, and (b) find the area of triangle PQR \(P(1,0,1),Q(-2,1,3), R(4,2,5)\)

The set \(B=\left\{1+t^{2}, t+t^{2}, 1+2t+t^{2}\right\}\) is a basis for \(\displaystyle{\mathbb{{{P}}}}_{{{2}}}\). Find the coordinate vector of \(\displaystyle{p}{\left({t}\right)}={1}+{4}{t}+{7}{t}^{{{2}}}\) relative to B.

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. \(P(3, 0, 1), Q(-1, 2, 5), R(5, 1, -1), S(0, 4, 2)\)

Use the cross product to find the sine of the angle between the vectors \(\displaystyle{u}={\left({2},{3},-{6}\right)}\) and \(v =(2,3,6)\).

Find \(\displaystyle\vec{{{a}}}+\vec{{{b}}},\ {2}\vec{{{a}}}+{3}\vec{{{b}}},\ {\left|\vec{{{a}}}\right|}\) and \(|\vec{a}-\vec{b}|\). \(\displaystyle\vec{{{a}}}={<}{5},-{12}{>},\ \vec{{{b}}}={<}-{3},-{6}{>}\)

The work W done by a constant force F in moving an object from a point A in space to a point B in space is defined as \(W=F \cdot \overrightarrow{AB}\). Find the work done by a force of 3 newtons acting in the direction \(\displaystyle{2}{i}+{j}+{2}{k}\) in moving an object 2 meters from \(\displaystyle{\left({0},{0},{0}\right)}\) to \(\displaystyle{\left({0},{2},{0}\right)}\).

Find the cross product \(a \times b\) and verify that it is orthogonal to both a and b. \(a=(2,3,0), b=(1,0,5)\)

Without calculation, find one eigenvalue and two linearly independent eigenvectors of \(A=\begin{bmatrix}5 & 5 & 5 \\5 & 5 & 5\\5 & 5 & 5 \end{bmatrix}\) Justify your answer.

Find the best approximation to z by vectors of the form \(\displaystyle{c}_{{1}}{v}_{{1}}+{c}_{{2}}{v}_{{2}}\) \(z=\begin{bmatrix}3\\-7\\2\\3\end{bmatrix},\ v_1=\begin{bmatrix}2\\-1\\-3\\1\end{bmatrix}\ v_2=\begin{bmatrix}1\\1\\0\\-1\end{bmatrix}\)