1. **State the problem:** We have two vectors $ \vec{G} $ and $ \vec{H} $ in 3D space with magnitudes and angles given. We need to find: (a) $ \vec{G} \times \vec{H} $ (the cross product), (b) $ |\vec{G} \times \vec{H}| $ (the magnitude of the cross product), (c) $ \vec{G} \cdot \vec{H} $ (the dot product). 2. **Given data:** - $ |\vec{G}| = 10.0 $ - $ |\vec{H}| = 15.0 $ - $ \vec{G} $ makes a 60° angle with the z-axis and its projection in the x-y plane makes a 45° angle with the x-axis. - $ \vec{H} $ makes a 45° angle with the z-axis and its projection in the x-y plane makes a 30° angle with the y-axis. 3. **Find components of $ \vec{G} $ and $ \vec{H} $:** For a vector with magnitude $ r $, angle $ \theta $ with z-axis, and projection angle $ \phi $ in x-y plane: $$ G_x = |G| \sin 60^\circ \cos 45^\circ $$ $$ G_y = |G| \sin 60^\circ \sin 45^\circ $$ $$ G_z = |G| \cos 60^\circ $$ Calculate: $$ G_x = 10 \times \sin 60^\circ \times \cos 45^\circ = 10 \times \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} = 10 \times \frac{\sqrt{6}}{4} = 2.5 \sqrt{6} \approx 6.1237 $$ $$ G_y = 10 \times \sin 60^\circ \times \sin 45^\circ = 10 \times \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} = 2.5 \sqrt{6} \approx 6.1237 $$ $$ G_z = 10 \times \cos 60^\circ = 10 \times \frac{1}{2} = 5 $$ Similarly for $ \vec{H} $: Projection angle is 30° with y-axis, so x and y components: $$ H_x = |H| \sin 45^\circ \sin 30^\circ $$ $$ H_y = |H| \sin 45^\circ \cos 30^\circ $$ $$ H_z = |H| \cos 45^\circ $$ Calculate: $$ H_x = 15 \times \sin 45^\circ \times \sin 30^\circ = 15 \times \frac{\sqrt{2}}{2} \times \frac{1}{2} = 15 \times \frac{\sqrt{2}}{4} = \frac{15 \sqrt{2}}{4} \approx 5.3033 $$ $$ H_y = 15 \times \sin 45^\circ \times \cos 30^\circ = 15 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = 15 \times \frac{\sqrt{6}}{4} = \frac{15 \sqrt{6}}{4} \approx 9.1849 $$ $$ H_z = 15 \times \cos 45^\circ = 15 \times \frac{\sqrt{2}}{2} = \frac{15 \sqrt{2}}{2} \approx 10.6066 $$ 4. **Calculate cross product $ \vec{G} \times \vec{H} $:** $$ \vec{G} \times \vec{H} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ G_x & G_y & G_z \\ H_x & H_y & H_z \end{vmatrix} $$ Calculate each component: $$ ( G_y H_z - G_z H_y ) \hat{i} = (6.1237 \times 10.6066 - 5 \times 9.1849) \hat{i} = (64.933 - 45.9245) \hat{i} = 19.0085 \hat{i} $$ $$ -( G_x H_z - G_z H_x ) \hat{j} = - (6.1237 \times 10.6066 - 5 \times 5.3033) \hat{j} = - (64.933 - 26.5165) \hat{j} = -38.4165 \hat{j} $$ $$ ( G_x H_y - G_y H_x ) \hat{k} = (6.1237 \times 9.1849 - 6.1237 \times 5.3033) \hat{k} = (56.255 - 32.466) \hat{k} = 23.789 \hat{k} $$ So: $$ \vec{G} \times \vec{H} = 19.0085 \hat{i} - 38.4165 \hat{j} + 23.789 \hat{k} $$ 5. **Calculate magnitude $ |\vec{G} \times \vec{H}| $:** $$ |\vec{G} \times \vec{H}| = \sqrt{19.0085^2 + (-38.4165)^2 + 23.789^2} $$ $$ = \sqrt{361.32 + 1475.88 + 566.01} = \sqrt{2403.21} \approx 49.02 $$ 6. **Calculate dot product $ \vec{G} \cdot \vec{H} $:** $$ \vec{G} \cdot \vec{H} = G_x H_x + G_y H_y + G_z H_z $$ $$ = 6.1237 \times 5.3033 + 6.1237 \times 9.1849 + 5 \times 10.6066 $$ $$ = 32.466 + 56.255 + 53.033 = 141.754 $$ **Final answers:** (a) $ \vec{G} \times \vec{H} = 19.01 \hat{i} - 38.42 \hat{j} + 23.79 \hat{k} $ (b) $ |\vec{G} \times \vec{H}| \approx 49.02 $ (c) $ \vec{G} \cdot \vec{H} \approx 141.75 $