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Worked solution

Calculate two angles in degrees and radians

Calculate the angle between a\mathbf a and b\mathbf b if (i) a=2i−3j+6k\mathbf a=2\mathbf i-3\mathbf j+6\mathbf k, b=i+j\mathbf b=\mathbf i+\mathbf j; (ii) a=2i−4j+5k\mathbf a=2\mathbf i-4\mathbf j+5\mathbf k, b=i+3j+8k\mathbf b=\mathbf i+3\mathbf j+8\mathbf k.

CJC · Vectors Basic · Self-Practice 3

(i) Negative cosine

Answer

1.671.67 rad (3 s.f.) or 95.8∘95.8^\circ (1 d.p.).

Compute all three components

a⋅b=2−3=−1,∣a∣=7,∣b∣=2,θ=cos⁡−1 ⁣(−172)≈1.67 rad≈95.8∘\mathbf a\cdot\mathbf b=2-3=-1,\quad|\mathbf a|=7,\quad|\mathbf b|=\sqrt2,\quad\theta=\cos^{-1}\!\left(-\frac1{7\sqrt2}\right)\approx1.67\text{ rad}\approx95.8^\circ

The missing k component of b is zero. Working is authored; both final forms are printed.

How did this part go?

(ii) Positive cosine

Answer

1.021.02 rad (3 s.f.) or 58.7∘58.7^\circ (1 d.p.).

Compute the product and both norms

a⋅b=2−12+40=30,∣a∣=45,∣b∣=74,θ=cos⁡−1 ⁣(304574)≈1.02 rad≈58.7∘.\begin{gathered}\mathbf a\cdot\mathbf b=2-12+40=30,\quad|\mathbf a|=\sqrt{45},\quad|\mathbf b|=\sqrt{74},\\\theta=\cos^{-1}\!\left(\frac{30}{\sqrt{45}\sqrt{74}}\right)\approx1.02\text{ rad}\approx58.7^\circ.\end{gathered}

Keep the inverse cosine in the calculator until final rounding.

How did this part go?

Solution notes

Both degrees/radians answers and their printed precision are retained; the source has no full working, so the derivations are authored.