Question

Given that a = 5i +2j - k and b = I - 3j + k. Find (a + b) x (a + b).

2i - 12j - 34k
2i + 12j + 34k
2i -3j + 12j
2i + 2k

Answer 1

Answer on Question #46883 – Math – Vector Calculus

1. Given that $\pmb{a} = 5\pmb{i} + 2\pmb{j} - \pmb{k}$ and $\pmb{b} = \pmb{i} - 3\pmb{j} + \pmb{k}$. Find $(\pmb{a} + \pmb{b}) \times (\pmb{a} + \pmb{b})$.

a) $2\mathbf{i} - 12\mathbf{j} - 34\mathbf{k}$

b) $2\mathbf{i} + 12\mathbf{j} + 34\mathbf{k}$

c) $2\mathbf{i} - 3\mathbf{j} + 12\mathbf{k}$

d) $2\mathbf{i} + 2\mathbf{k}$

**Remark.**

We know, that cross product of identical vectors equal zero. For example, $\pmb{n} \times \pmb{n} = 0$. In our case we have the same situation. But we have no option "zero". I think the condition of question is a little wrong — it must be: find $(a + b) \times (a - b)$.

**Solution.**

We know, that cross product $\pmb{n} \times \pmb{m} = (n_y m_z - n_z m_y)\pmb{i} + (n_z m_x - n_x m_z)\pmb{j} +$

$$+ (n _ {x} m _ {y} - n _ {y} m _ {x}) \pmb {k}.$$

Now we must find $(a + b)$ and $(a - b)$:

$$\pmb {a} + \pmb {b} = (5 + 1) \pmb {i} + (2 - 3) \pmb {j} + (- 1 + 1) \pmb {k} = 6 \pmb {i} - \pmb {j};$$ $$\pmb {a} - \pmb {b} = (5 - 1) \pmb {i} + (2 + 3) \pmb {j} + (- 1 - 1) \pmb {k} = 4 \pmb {i} + 5 \pmb {j} - 2 \pmb {k}.$$

And then we can find $(a + b) \times (a - b)$:

$$\begin{array}{l} (\pmb {a} + \pmb {b}) \times (\pmb {a} - \pmb {b}) = \left| \begin{array}{ccc} \pmb {i} & \pmb {j} & \pmb {k} \\ 6 & - 1 & 0 \\ 4 & 5 & - 2 \end{array} \right| = \\ = \left((-1)(-2) - 0(5)\right)\mathbf{i} + (0 \cdot 4 - 6(-2))\mathbf{j} + (6 \cdot 5 - (-1)4)\mathbf{k} = \\ \end{array}$$ $$= 2\mathbf{i} + 12\mathbf{j} + 34\mathbf{k}.$$

**Answer:**

b) $2\mathbf{i} + 12\mathbf{j} + 34\mathbf{k}$ is correct.

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