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Q16E

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Found in: Page 564

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

# (a) Find the magnitude of cross product$$|a \times b|$$.(b) Check whether the components of $$a \times b$$ are positive, negative or 0.

(a) The value of the cross product $$|a \times b|$$ is 6.

(b) The $$x$$ component of $${\rm{a}} \times {\rm{b}}$$ is a positive, $$y$$ component of $${\rm{a}} \times {\rm{b}}$$ is a negative, and $$z$$ component of $${\rm{a}} \times {\rm{b}}$$ is zero.

See the step by step solution

## Step 1: Formula used

If $$\theta$$ is the angle between vectors $${\bf{u}}$$ and $${\bf{v}}$$, then the cross product is,

$$|u \times v| = |u||v|\sin \theta \ldots \ldots \ldots (1)$$

Here,

|u| is the magnitude of vector$${\bf{u}}$$ and

|v| is the magnitude of vector$${\bf{v}}$$.

Right-hand rule:

The fingers of the right hand curl in the direction of rotation from $$a$$ to $$b$$, then the thumb points are in the direction of $$a \times b$$. The rotation angle should be less than$${180^^\circ }$$.

## Step 2: Find the $$|a \times b|$$

(a)

As given, $$|{\rm{a}}| = 3,|\;{\rm{b}}| = 2$$.

So, the value of angle $$(\theta )$$ is $${90^^\circ }$$by applying the right hand rule

Substitute 3 for $$|{\rm{a}}|,2$$ for $$|{\rm{b}}|$$, and $${90^^\circ }$$ for $$\theta$$ in $$|{\rm{a}} \times {\rm{b}}| = |{\rm{a}}||{\rm{b}}|\sin \theta$$

$$\begin{array}{l}|a \times b| = (3)(2)\sin \left( {{{90}^^\circ }} \right)\\|a \times b| = 6(1)\\|a \times b| = 6\end{array}$$

Thus, the magnitude of cross product $$|a \times b|$$ is 6.

## Step 3: Check whether the components of $$a \times b$$ are positive, negative or 0

(b)

From part (a), it is observed that, as vector a lies on the x y- plane, the $$z$$-component of $$a \times b$$is zero.

When the direction of vector $${\rm{a}} \times {\rm{b}}$$ is opposite to the direction of $$y$$ axis, the $$y$$-component of $${\rm{a}} \times {\rm{b}}$$ is negative.

As the direction of vector $${\rm{a}} \times {\rm{b}}$$ is in the direction of $$x$$ axis, the $$x$$-component of $${\rm{a}} \times {\rm{b}}$$ is positive.

Therefore, the $$x$$ component of $${\rm{a}} \times {\rm{b}}$$ is a positive, $$y$$ component of $${\rm{a}} \times {\rm{b}}$$ is a negative, and $$z$$ component of $$a \times b$$ is zero.