 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2021-resit/3/a/solution

n_{A} = (1, 2, - 4)

and

n_{B} = (2, - 1, 3)

. Their dot product is

- 12

, and their norms are

21

and

14

. The usual angle between two planes is the smaller, unoriented angle, so

θ = arccos \frac{12}{21 14} = arccos \frac{2 6}{7} \approx 45. 6^{\circ} .

For the particular signed normal vectors printed above, the angle is the supplement

π - θ \approx 134. 4^{\circ}

. Reversing either normal vector exchanges these two angles without changing either plane.

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