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Solve the following.
Let α⃗ = 3î + ĵ and β⃗ = 2î - ĵ + 3k̂. If β⃗ = β⃗₁ - β⃗₂ where β⃗₁ is parallel to α⃗ and β⃗₂ is…
Let $\vec{\alpha} = 3\hat{i} + \hat{j}$ and $\vec{\beta} = 2\hat{i} - \hat{j} + 3\hat{k}$. If $\vec{\beta} = \vec{\beta}_1 - \vec{\beta}_2$ where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$.
(a)[1]
Find $\vec{\beta}_1$.
(b)[1.5]
Find $\vec{\beta}_2$.
(c)[1.5]
Hence, find $\vec{\beta}_1 \times \vec{\beta}_2$.
Answer
Answer (a)
Official answer key$\vec{\beta}_1 = \frac{3}{2}\hat{i} + \frac{1}{2}\hat{j}$
Final answer: $\frac{3}{2}\hat{i} + \frac{1}{2}\hat{j}$
Answer (b)
Official answer key$\vec{\beta}_2 = -\frac{1}{2}\hat{i} + \frac{3}{2}\hat{j} - 3\hat{k}$
Final answer: $-\frac{1}{2}\hat{i} + \frac{3}{2}\hat{j} - 3\hat{k}$
Answer (c)
Official answer key$\vec{\beta}_1 \times \vec{\beta}_2 = -\frac{3}{2}\hat{i} + \frac{9}{2}\hat{j} + \frac{5}{2}\hat{k}$
Final answer: $-\frac{3}{2}\hat{i} + \frac{9}{2}\hat{j} + \frac{5}{2}\hat{k}$