Question

for the vectors in the figure, find the following.
(a) $(vec{a} \times \vec{f}) \cdot \vec{d}$
(b) $(vec{a} \times \vec{f}) \cdot (\vec{d} \times \vec{b})$
(c) $(vec{a} \cdot \vec{f})(\vec{d} \times \vec{b})$ (express your answer in vector form.)
$(vec{a} \cdot \vec{f})(\vec{d} \times \vec{b}) = $

Explanation:

First, we define each vector in component form using their magnitudes and angles:

Step1: Define vector $\vec{A}$

$\vec{A} = 10.2\cos30^\circ\hat{i} + 10.2\sin30^\circ\hat{j} = 8.83\hat{i} + 5.1\hat{j}$

Step2: Define vector $\vec{B}$

$\vec{B} = 5.4\cos53^\circ\hat{i} + 5.4\sin53^\circ\hat{j} = 3.25\hat{i} + 4.31\hat{j}$

Step3: Define vector $\vec{D}$

$\vec{D} = 20.8\cos(180^\circ-37^\circ)\hat{i} + 20.8\sin(180^\circ-37^\circ)\hat{j} = -16.61\hat{i} + 12.51\hat{j}$

Step4: Define vector $\vec{F}$

$\vec{F} = 20.8\cos(180^\circ+30^\circ)\hat{i} + 20.8\sin(180^\circ+30^\circ)\hat{j} = -18.01\hat{i} - 10.4\hat{j}$

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Part (a): $(\vec{A} \times \vec{F}) \cdot \vec{D}$

Step5: Compute $\vec{A} \times \vec{F}$

For 2D vectors, $\vec{U} \times \vec{V} = (U_xV_y - U_yV_x)\hat{k}$
$\vec{A} \times \vec{F} = (8.83(-10.4) - 5.1(-18.01))\hat{k} = (-91.83 + 91.85)\hat{k} \approx 0\hat{k}$

Step6: Compute dot product with $\vec{D}$

$(\vec{A} \times \vec{F}) \cdot \vec{D} = 0\hat{k} \cdot (-16.61\hat{i} + 12.51\hat{j}) = 0$

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Part (b): $(\vec{A} \times \vec{F}) \cdot (\vec{D} \times \vec{B})$

Step7: Compute $\vec{D} \times \vec{B}$

$\vec{D} \times \vec{B} = (-16.61(4.31) - 12.51(3.25))\hat{k} = (-71.59 - 40.66)\hat{k} = -112.25\hat{k}$

Step8: Compute dot product

$(\vec{A} \times \vec{F}) \cdot (\vec{D} \times \vec{B}) = 0\hat{k} \cdot (-112.25\hat{k}) = 0$

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Part (c): $(\vec{A} \cdot \vec{F})(\vec{D} \times \vec{B})$

Step9: Compute $\vec{A} \cdot \vec{F}$

$\vec{A} \cdot \vec{F} = (8.83)(-18.01) + (5.1)(-10.4) = -159.03 - 53.04 = -212.07$

Step10: Multiply by $\vec{D} \times \vec{B}$

$(\vec{A} \cdot \vec{F})(\vec{D} \times \vec{B}) = -212.07(-112.25\hat{k}) = 23805\hat{k}$

Answer:

(a) $0$
(b) $0$
(c) $\boldsymbol{23805\hat{k}}$