Question

Let \[ \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}, \quad \vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}, \quad \vec{c} = c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k}, \] show that \[ \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}. \]

Hint:

The cross product of vectors is linear, which means it distributes over addition.

Answer 1

Step 1: Use the distributive property of the cross product: \[ \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} \] Step 2: The cross product is linear, meaning it satisfies the distributive property. Therefore, the equation holds as shown above. Thus, \( \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} \).