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Rom4ik [11]
3 years ago
5

Find the angle between the vectors. (first find an exact expression and then approximate to the nearest degree.) a = 1, −4, 1 ,

b = 0, 3, −3
Mathematics
1 answer:
qaws [65]3 years ago
5 0

The formula of the dot product of two vectors:

\overrightarrow{a}\circ\overrightarrow{b}=|\overrightarrow{a}|\cdot|\overrightarrow{b}|\cos\alpha\to\cos\alpha=\dfrac{\overrightarrow{a}\circ\overrightarrow{b}}{|\overrightarrow{a}|\cdot|\overrightarrow{b}|}

We have:

\overrightarrow{a}=[1,\ -4,\ 1]\\\\\overrightarrow{b}=[0,\ 3,\ -3]

Calculate the dot product:

\overrightarrow{a}\circ\overrightarrow{b}=1\cdot0+(-4)\cdot3+1\cdot(-3)=-15

Calculate the lengths of vectors:

|\overrightarrow{a}|=\sqrt{1^2+(-4)^2+1^2}=\sqrt{18}=\sqrt{9\cdot2}=\sqrt9\cdot\sqrt2=3\sqrt2\\\\|\overrightarrow{b}|=\sqrt{0^2+3^2+(-3)^2}=\sqrt{18}=\sqrt{9\cdot2}=\sqrt9\cdot\sqrt2=3\sqrt2

Substitute:

\cos\alpha=\dfrac{-15}{3\sqrt2\cdot3\sqrt2}=\dfrac{-15}{9\sqrt4}=-\dfrac{15}{18}\approx-0.8333\to\alpha\approx146^o

Answer: Measure of the angle between the vectors a and b is equal 146°.

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