Question

Let v (3, 0, 1) and w (3,-1,-4). Compute the following quantities:

a. Calculate the parallel projection of v onto w

Answer 1

The parallel projection of vector v onto vector w is calculated using the projection formula, resulting in (0.577, -0.192, -0.769).

Step-by-step explanation:

The question asks for the calculation of the parallel projection of vector v onto vector w. To find the parallel projection of vector v onto vector w, we use the formula for the projection of v onto w, which is given by:

projwv = (v · w / |w|2) × w

First, calculate the dot product of vectors v and w:

v · w = (3)(3) + (0)(-1) + (1)(-4) = 9 + 0 - 4 = 5

Next, find the magnitude squared of vector w:

|w|2 = (3)2 + (-1)2 + (-4)2 = 9 + 1 + 16 = 26

Then, multiply the scalar (v · w / |w|2) with vector w to get the projection.

projwv = (5 / 26) × (3, -1, -4)

= (15/26, -5/26, -20/26)

Using decimal notation, projwv = (0.577, -0.192, -0.769)