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2 problems match “vector norm”. Clear

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1. Cosine of angle between vectors ●●○○○ 20 ptsFor certain vectors $\mathbf{p}$ and $\mathbf{q}$, the vectors $3 \mathbf{p} + \mathbf{q}$ and $5 \mathbf{p} - 3 \mathbf{q}$ are orthogonal. Also, the vectors $2 \mathbf{p} + \mathbf{q}$ and $4 \mathbf{p} - 2 \mathbf{q}$ are orthogonal. If $\theta$ is the angle between $\mathbf{p}$ and $\mathbf{q}$, find $\cos \theta$. Give your answer as a decimal.
2. Dot product from norms and cross product ●●○○○ 20 ptsLet $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 5,$ and $\|\mathbf{a} \times \mathbf{b}\| = 8.$ Find $|\mathbf{a} \cdot \mathbf{b}|.$