Question

Let a = 〈 − 2 , − 1 〉 and b = 〈 − 8 , − 9 〉 . Compute the projection of "a" onto "b" and the vector component of "a" orthogonal to "b" .

Answer 1

To compute the projection of vector a onto vector b, use the formula proj_b a = (a · b) / |b| * (b / |b|). To find the vector component of a orthogonal to b, subtract the projection of a onto b from a.

Step-by-step explanation:

To compute the projection of vector a onto vector b, we can use the formula:

projba = (a · b) / |b| * (b / |b|)

where · represents the dot product, |b| represents the magnitude of b, and / represents division. Let's calculate it step by step:

1. Calculate the dot product: a · b = (-2) * (-8) + (-1) * (-9) = 16 - 9 = 7
2. Calculate the magnitude of b: |b| = sqrt((-8)^2 + (-9)^2) = sqrt(64 + 81) = sqrt(145)
3. Calculate the unit vector in the direction of b: b / |b| = (-8) / sqrt(145)î + (-9) / sqrt(145)ĵ
4. Multiply the dot product by the unit vector: projba = 7 * ((-8) / sqrt(145)î + (-9) / sqrt(145)ĵ)

Therefore, the projection of a onto b is projba = (7 * (-8)) / sqrt(145)î + (7 * (-9)) / sqrt(145)ĵ. To find the vector component of a orthogonal to b, we can subtract the projection of a onto b from a:

compba = a - projba

So, the vector component of a orthogonal to b is compba = (-2, -1) - ((7 * (-8)) / sqrt(145), (7 * (-9)) / sqrt(145)).