Question

In Exercises \(1-6,\) find \((\mathbf{a}) \mathbf{u} \cdot \mathbf{v},(\mathbf{b}) \mathbf{u} \cdot \mathbf{u},(\mathbf{c})\|\mathbf{u}\|^{2},(\mathbf{d})(\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\) and \((e) u \cdot(2 v)\). $$ \mathbf{u}=\langle 5,-1\rangle, \quad \mathbf{v}=\langle-3,2\rangle $$

Short answer

The solutions are: a) -17, b) 26, c) 26, d) 51i - 34j, and e) -34.

Step-by-step solution

Calculate the Dot Product of u and v

The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is calculated as follows: \( \mathbf{u} \cdot \mathbf{v} = (5) * (-3) + (-1) * (2) = -15 - 2 = -17.\)

Calculate the Dot Product of u and u

The dot product of \(\mathbf{u}\) with itself is calculated as follows: \( \mathbf{u} \cdot \mathbf{u} = (5) * (5) + (-1) * (-1) = 25 + 1 = 26.\)

Calculate the Square of the Vector Norm

The square of the norm of \(\mathbf{u}\) is equal to the dot product of \(\mathbf{u}\) with itself, which we calculated previously to be 26.

Calculate the Scaled Vector

The result of dot product between \(\mathbf{u}\) and \(\mathbf{v}\) is -17. Scale \(\mathbf{v}\) by this quantity, gives a vector \( ( -17 * -3, -17 * 2) = (51, -34).\)

Calculate the Dot Product of u and Scaled v

Multiply \(\mathbf{v}\) by 2 to get a new vector, then calculate its dot product with \(\mathbf{u}\): \(\mathbf{u} \cdot (2\mathbf{v}) = (5) * (2*-3) + (-1) * (2*2) = -30 - 4 = -34.\)