8. DETAILS None of the parts below are related to each other. (a) Find a vector perpendicular to both \(\vec{v} = 2\vec{i} + 3\vec{k}\) and \(\vec{w} = 4\vec{j} + 5\vec{k}\). (b) Find a vector of length 2 that makes an angle of \(\frac{3\pi}{4}\) with the positive x-axis. (b) Suppose \(\vec{a} \cdot \vec{b} = 11\) where \(\vec{a}\) is a unit vector. If the cosine of the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{1}{5}\), find \(||\vec{b}|| = \boxed{\hspace{1cm}})
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To find a vector perpendicular to two given vectors, we can take their cross product. Let's denote the first vector as A = <2, 7, 3k> and the second vector as B = <4, 5, 0>. To find the cross product of A and B, we can use the formula: A x B = <(AyBz - AzBy), Show more…
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$$\begin{array}{l}{\text { In Exercises } 1-8, \text { find }} \\ {\text { a. } \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|} \\ {\text { b. the cosine of the angle between } \mathbf{v} \text { and } \mathbf{u}} \\ {\text { c. the scalar component of } \mathbf{u} \text { in the direction of } \mathbf{v}} \\ {\text { d. the vector proj, u. }}\end{array}$$ $$\mathbf{v}=5 \mathbf{i}+\mathbf{j}, \quad \mathbf{u}=2 \mathbf{i}+\sqrt{17} \mathbf{j}$$
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