Question

Find the angle between the vectors a = (1, -4, 1) and b = (0, 9, -9). First, find an exact expression and then approximate to the nearest degree.

Answer 1

To find the angle between vectors a and b, calculate the dot product, find the magnitudes of both vectors, then use these to compute the cosine of the angle, and finally take the inverse cosine to find the angle itself.

Step-by-step explanation:

To find the angle between two vectors, we use the dot product formula and the magnitudes of the vectors. For vectors a = (1, -4, 1) and b = (0, 9, -9), the angle θ can be found using the following steps:

1. Calculate the dot product of vectors a and b: a · b = (1)(0) + (-4)(9) + (1)(-9) = -36 - 9 = -45.
2. Find the magnitudes of a and b: |a| = √(12 + (-4)2 + 12) = √(1 + 16 + 1) = √18, and |b| = √(02 + 92 + (-9)2) = √(0 + 81 + 81) = √162.
3. Use the dot product and magnitudes to find the cosine of the angle: cos(θ) = (a · b) / (|a| |b|) = -45 / (√18 √162).
4. Calculate the angle θ by taking the inverse cosine: θ = cos-1(-45 / (√18 √162)).
5. Approximate θ to the nearest degree using a calculator.

Using these steps, the student can find both the exact and approximate angles between the vectors a and b.

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