Can someone start me off on these and please explain step by step how you got everything because I'm having difficulty understanding the entire concept.

Find the projection of u onto v and calculate its magnitude.

a) u = (2,5) v = (6,4)
b) u = (3, 6, -2) v = (-4, 3, 8)

2) If u and v are non zero vectors, but Projection (u unto v) = 0, what conclusion can be drawn?
Would that also mean Projection (v unto u) = 0?

3) Find the projection of PQ onto each of the coordinate axes, where point (2,3,5) and Q is the point (-1,2,5).

The projection of vector u onto vector v is defined as

(u∙v)/│v│

u∙v = (2,5)∙(6,4) = 12+20 = 32
│v│ = √(6^2+4^2) = √52

so the projection of u onto v
= 32/√52 = 16/√13

follow this method for the other questions.

Think of the "projection of u onto v" as the 'shadow' cast by u onto v by a light from above shining perpendicular to v.

So if u does not cast a shadow on v, (the projection is zero), what should that tell you about the direction of u in relation to v ??

Sure, let's go through each question step by step and explain the concepts involved.

1) Find the projection of u onto v and calculate its magnitude.

a) u = (2,5) v = (6,4)
To find the projection of u onto v, we first need to compute the dot product of u and v, and then divide it by the magnitude of v squared.

- Dot product of u and v: u.v = (2*6) + (5*4) = 12 + 20 = 32
- Magnitude of v squared: |v|^2 = (6^2) + (4^2) = 36 + 16 = 52

Now we can calculate the projection of u onto v:
Projection of u onto v = (u.v / |v|^2) * v = (32 / 52) * (6,4) = (12/13) * (6,4) = (72/13, 48/13)

To calculate the magnitude of the projection, we can use the formula:
Magnitude of projection of u onto v = sqrt((72/13)^2 + (48/13)^2) = sqrt(5184/169 + 2304/169) = sqrt(7488/169) = sqrt(44) = 2√11

b) u = (3, 6, -2) v = (-4, 3, 8)
Similarly, we first compute the dot product of u and v, and then divide it by the magnitude of v squared.

- Dot product of u and v: u.v = (3*-4) + (6*3) + (-2*8) = -12 + 18 - 16 = -10
- Magnitude of v squared: |v|^2 = (-4^2) + (3^2) + (8^2) = 16 + 9 + 64 = 89

Now we can calculate the projection of u onto v:
Projection of u onto v = (u.v / |v|^2) * v = (-10 / 89) * (-4, 3, 8) = (40/89, -30/89, -80/89)

To calculate the magnitude of the projection, we use the formula:
Magnitude of projection of u onto v = sqrt((40/89)^2 + (-30/89)^2 + (-80/89)^2) = sqrt(1600/7921 + 900/7921 + 6400/7921) = sqrt(8900/7921) = sqrt(100) = 10

2) If Projection (u unto v) = 0, what conclusion can be drawn? Would that also mean Projection (v unto u) = 0?

If the projection of u onto v equals zero, it means that u and v are orthogonal (perpendicular) to each other. In other words, u and v are at a 90-degree angle to each other.

However, this does not necessarily mean that the projection of v onto u is also zero. The projection of v onto u can still have a non-zero value. Orthogonality is a symmetric relationship, meaning if u is orthogonal to v, then v is also orthogonal to u, but the specific values of the projections can be different.

3) Find the projection of PQ onto each of the coordinate axes, where point P is (2,3,5) and Q is (-1,2,5).
To find the projection of PQ onto each coordinate axis, we can find the components of PQ along each axis.

The x-coordinate of PQ = -1 - 2 = -3
The y-coordinate of PQ = 2 - 3 = -1
The z-coordinate of PQ = 5 - 5 = 0

Therefore, the projection of PQ onto the x-axis is (-3, 0, 0).
The projection of PQ onto the y-axis is (0, -1, 0).
The projection of PQ onto the z-axis is (0, 0, 0).

Note that the projection onto the z-axis is zero because the change in the z-coordinate is zero, indicating that PQ lies in the x-y plane.