21. What can you conclude about the angle between two non-zero vectors $$ \boldsymbol{u} $$ and $$ \boldsymbol{v} $$ if $$ \boldsymbol{u} \cdot \boldsymbol{v}=|\boldsymbol{u} imes \boldsymbol{v}| $$ ?

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21. What can you conclude about the angle between two non-zero vectors $$ \boldsymbol{u} $$ and $$ \boldsymbol{v} $$ if $$ \boldsymbol{u} \cdot \boldsymbol{v}=|\boldsymbol{u} 	imes \boldsymbol{v}| $$ ?

UpStudy AI · Accepted Answer

We start with the formulas for the dot product and the magnitude of the cross product for two non-zero vectors $$ \boldsymbol{u} $$ and $$ \boldsymbol{v} $$:

$$
\boldsymbol{u} \cdot \boldsymbol{v} = |\boldsymbol{u}| \, |\boldsymbol{v}| \cos\theta
$$

$$
|\boldsymbol{u} \times \boldsymbol{v}| = |\boldsymbol{u}| \, |\boldsymbol{v}| \sin\theta
$$

Given that

$$
\boldsymbol{u} \cdot \boldsymbol{v} = |\boldsymbol{u} \times \boldsymbol{v}|,
$$

we substitute the expressions above:

$$
|\boldsymbol{u}| \, |\boldsymbol{v}| \cos\theta = |\boldsymbol{u}| \, |\boldsymbol{v}| \sin\theta.
$$

Since $$ |\boldsymbol{u}| $$ and $$ |\boldsymbol{v}| $$ are non-zero, we can divide both sides by $$ |\boldsymbol{u}| \, |\boldsymbol{v}| $$, yielding:

$$
\cos\theta = \sin\theta.
$$

To solve for $$ \theta $$, we divide both sides by $$ \cos\theta $$ (assuming $$ \cos\theta \neq 0 $$):

$$
1 = \tan\theta.
$$

This implies that:

$$
\tan\theta = 1.
$$

The angle $$ \theta $$ that satisfies this equation (with $$ 0 \leq \theta \leq \pi $$) is:

$$
\theta = \frac{\pi}{4} \quad \text{(or } 45^\circ\text{)}.
$$
The angle between the two vectors is $$ 45^\circ $$.

What can you conclude about the angle between two non-zero vectors and if ?

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