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The vectors u and v are such that $|u|=4$ $|v|=5$ $u \cdot (u + v) = 21$ - Scottish Highers Maths - Question 14 - 2019

Question 14

$The-vectors-u-and-v-are-such-that--$|u|=4$--$|v|=5$--$u-\cdot-(u-+-v)-=-21$-Scottish Highers Maths-Question 14-2019.png$

The vectors u and v are such that $|u|=4$ $|v|=5$ $u \cdot (u + v) = 21$. Determine the size of the angle between the vectors u and v.

Worked Solution & Example Answer:The vectors u and v are such that $|u|=4$ $|v|=5$ $u \cdot (u + v) = 21$ - Scottish Highers Maths - Question 14 - 2019

Step 1

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Answer

Using the formula for dot product:

$u \cdot (u + v) = u \cdot u + u \cdot v$

We know that:

So we can say:

$u \cdot v = 21 - 16 = 5$

Step 2

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Answer

We know that:

$u \cdot v = |u| |v| \cos \theta$

Substituting the values we have:

$5 = 4 \cdot 5 \cos \theta$

Thus:

$5 = 20 \cos \theta$

Therefore,

$\cos \theta = \frac{5}{20} = \frac{1}{4}$

Step 3

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Answer

To find $\theta$ , we take the arccosine of both sides:

$\theta = \cos^{-1}\left(\frac{1}{4}\right)$

Calculating this will give:

$\theta \approx 75.5^\circ \, \text{or} \, 1.31 \, \text{radians}$

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