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Find the angle between the vectors
\(\underset{\sim}{a}=\underset{\sim}{i}+2 \underset{\sim}{j}-3 \underset{\sim}{k}\)
\(\underset{\sim}{b}=-\underset{\sim}{i}+4 \underset{\sim}{j}+2 \underset{\sim}{k}\),
giving your answer to the nearest degree. (3 marks)
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\(87^{\circ} \)
\[\underset{\sim}{a}=\left(\begin{array}{c} 1 \\ 2 \\ -3 \end{array}\right),\ \ \underset{\sim}{b}=\left(\begin{array}{c} -1 \\ 4 \\ 2 \end{array}\right) \]
\(\Big{|} \underset{\sim}{a} \Big{|} = \sqrt{1+4+9} = \sqrt{14} \)
\(\Big{|} \underset{\sim}{b} \Big{|} = \sqrt{1+16+4} = \sqrt{21} \)
\( \underset{\sim}{a} \cdot \underset{\sim}{b} = -1 + 8-6=1 \)
| \(\cos\ \theta \) | \(=\dfrac{\underset{\sim}{a} \cdot \underset{\sim}{b}}{\Big{|}\underset{\sim}{a}\Big{|} \cdot \Big{|}\underset{\sim}{b}\Big{|}} \) | |
| \(=\dfrac{1}{\sqrt{294}} \) | ||
| \( \theta\) | \(=\cos ^{-1} \Big{(}\dfrac{1}{\sqrt{294}}\Big{)} \) | |
| \(=86.65…\) | ||
| \(=87^{\circ} \) |