Question

Use a computer algebra system to find \(\mathbf{u} \times \mathbf{v}\) and a unit vector orthogonal to \(\mathbf{u}\) and \(\mathbf{v}\). $$ \begin{array}{l} \mathbf{u}=-3 \mathbf{i}+2 \mathbf{j}-5 \mathbf{k} \\ \mathbf{v}=\frac{1}{2} \mathbf{i}-\frac{3}{4} \mathbf{j}+\frac{1}{10} \mathbf{k} \end{array} $$

Short answer

The cross product \(\mathbf{u} \times \mathbf{v}\) is \(\frac{-7}{10}i - \frac{11}{10}j - \frac{11}{10}k\). The required unit vector is \(\approx 0.39i + 0.55j + 0.55k\)

Step-by-step solution

Compute the Cross Product

The cross product \(\mathbf{u} \times \mathbf{v}\) can be found using the determinant of the following 3x3 matrix. The first row is with unit vectors (i, j, k), the second row with the components of \(\mathbf{u}\) (-3, 2, -5) and the third row with the components of \(\mathbf{v}\) (1/2, -3/4, 1/10). After evaluating the determinant, the resulting vector is \(-3*\frac{1}{10}+2*\frac{3}{4}+\frac{1}{2}*5-(-3)*\frac{3}{4}+2*\frac{1}{10}\) for i-component, \(-3*\frac{1}{10}-(-5)*\frac{1}{2}+(-3)*\frac{1}{10}-5*\frac{1}{2}\) for j-component, \(-3*(-3/4)+2*\frac{1}{2}-(-3)*\frac{1}{2}+2*\frac{1}{10}\) for k-component.

Simplify the Cross Product

Simplify the expressions for each component separately. After simplification, the resulting vector is \(\frac{-7}{10}i - \frac{11}{10}j - \frac{11}{10}k\)

Normalize the Cross Product to Find the Unit Vector

The unit vector is the vector obtained by dividing each component of \(\mathbf{u} \times \mathbf{v}\) by its magnitude. The magnitude can be found as follows: \(\sqrt{(-7/10)^2 + (-11/10)^2 + (-11/10)^2}\). Divide each component of \(\mathbf{u} \times \mathbf{v}\) by this magnitude to get the unit vector.

Simplify the Unit Vector

Evaluate the magnitude and divide each component to get the unit vector. The unit vector obtained is approximately \(0.39i + 0.55j + 0.55k\)