Question

Find the angle between the given vectors . Round to the nearest tenth of a degree. 1) u= -3i +6j ,v=5i + 2j 2) u= i -j , v=2i +3j Use the dot product

Vectors
ANSWERED
asked 2021-06-02
Find the angle between the given vectors . Round to the nearest tenth of a degree.
1) u= -3i +6j ,v=5i + 2j
2) u= i -j , v=2i +3j
Use the dot product to determiine wheter the vectors are parallel, orthogonal , or neither .
1) v = 2i +j, w = i-2j
2) v = 4i-j , w=8i-2j
3) v= 3i +3j , w=3i -2j
Find proj w v
1) v = 2i +3j . w = 8i- 6j
2) v= 2i -3j . w =-3i +j

Expert Answers (2)

2021-06-03
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Best answer
2021-09-08

1) \(4=-3i+6j;\ v=5i+2j\)

\(\cos\theta=\frac{u\cdot v}{|u||v|}=\frac{(-3i+6j)(5i+2j)}{|(-3i+6j)|\cdot|(5i+2j)|}\)

\(\to\cos\theta=\frac{-15+12}{\sqrt{9+36}\cdot\sqrt{25+4}}=\frac{-3}{3\sqrt{5}\cdot\sqrt{29}}=\frac{-1}{\sqrt{145}}\)

\(\theta=\cos^{-1}(\frac{-1}{\sqrt{145}})=\pi-\cos^{-1}(\frac{1}{\sqrt{145}})=180-88-63^\circ\)

\(\theta=91.372\)

2) \(v=2i+j,\ w=i-ej\)

\(\to v\cdot w=(2i+j)(i-2j)=2-2=0\)

\(v\cdot w=0\Rightarrow\) V and w are orthogonal

3) \(\text{proj}_wv=\frac{w\cdot v}{v\cdot v}\cdot v\)

\(\text{proj}_wv=\frac{(2i+3j)(8i-6j)}{(2i+3j)(2i+3j)}(2i+3j)\)

\(=\frac{16-18}{4+9}(2i+3j)=-\frac{2}{13}(2i+13j)\)

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