Vector help

The position vectors of points A and B are (7, 1, -3) and (-2,8,-1)

a) Calculate vector OA . vector OB

Very simple, (7, 1, -3) . (-2, 8, -1) = -14 + 8 + 3 = -3

b) Find the size, in degrees, of the acute angle between the vectors.

lal = root (49 + 1 + 9) = root 59

lbl = root (4 + 64 + 1) = root 69

a.b = lal lbl cos x
so, -3 = root 59 x root 69 (cos x)
so -3 = root (4071) cos x
so cos x = -3/root (4071)

x = arccos (-3/root (4071)
= 92.7 degrees,

but it says the acute angle, so the angle = 180-x = 87.3

But the textbook says the answer is 87.1? Have I done anything wrong?

= 92.7 degrees,

but it says the acute angle, so the angle = 180-x = 87.3

But the textbook says the answer is 87.1? Have I done anything wrong?

I need help with another question as I disagree with their answer, this one i'm less confident with my workings:

Is the point (-2,4,7) closer to the line with equation r = 5i + 2k + λ(i - 2j + 4k) or the line parallel to the vector 2i + j that passes through the origin?:

we know Line 1: r = (5, 0, 2) + λ (1, -2, 4)
and line 2: r = (0,0,0) + s (2, 1, 0)

let point B be on line 1 so that the vector AB is perpendicular to line 1.

so vector AB . (1, -2, 4) = 0

Vector OA = (-2, 4, 7) and Vector OB = (5 + λ, -2λ, 2 + 4λ)

so vector AB = (7 + λ, -2λ - 4, 4λ - 5)

so (7 + λ, -2λ - 4, 4λ - 5) . (1, -2, 4) = 0
7 + λ - 2(-2λ - 4) + 4 (4λ - 5) = 0
7 + λ + 4 λ + 8 + 16λ - 20 = 0
so λ = 5/21

so Vector AB = (152/21, -94/21, -85/21)

lABl = root (1865/21) = 9.424 (3sf)

Let point C be on line 2 so that vector AC is perpendicular to line 2

So vector OC = (2s, s, 0)

vector AC = (2s, s, 0) - (-2, 4, 7)
= (2s + 2, s - 4, -7)

So Vector AC . (2, 1, 0) = 0

(2s + 2, s - 4, -7) . (2, 1, 0) = 0

4s + 4 + s -4 = 0
5s = 0
so s = 0

so Vector AC = (2, -4, -7)

lACl = root 69

therefore, the point is closer to the second line.

However, the book says it's closer to the first?

Have I gone wrong or is it another textbook mistake?

This is where I struggle to visualise it, I would try and use dot product but I can't figure out how the angle calculated plays into it, it could potentially be 180 - the angle from dot product and then simply 1/2 ab sin C to find the area.

So I'm not too sure on what to do, or how to visualise it. I also tried a sketch but I couldn't seem to display the vectors.

I also worked out lOAl as root 186
and lOBl as 3 root 6.

A simpler question:

This is for my mate but I wasn't able to explain it for him:

Points A and B have position vectors OA = (2, 2, 3) and OB = (-1, 7, 2). Find the angle between AB and OA

The textbook specifies to use the scalar product of two vectors to find the angle between them, you need both vectors to be directed away from the angle, otherwise the angle you find will be 180 - the angle.

His workings:

Let the angle between the position vectors AB and AO be x.

Vector AB = (-1, 7, 2) - (2, 2, 3) = (-3, 5, -1)
lABl = root 35
A
Vector AO = (-2, -2, -3). lAOl = root 17

AB . AO = -1

x = arccos (-1/root 595) = 92.3 degrees

However, the answer given is 87.7 degrees.

This has confused both me and him, have they calculated the wrong angle? And if not why is it 180 - x

Nothing in the question suggests you seek the acute/obtuse angle, so no need to aim for one or the other.

Also, since the questions says between AB and OA then those are the vectors you need to dot.
You dotted AB with AO instead.

The dot product formula gives the angle between the vectors that are ‘head-to-head’ or ‘tail-to-tail’. It may not seem that you can take the dot product between AB and OA to get anything meaningful given this, but notice that if (on a diagram) you extend the vector OA through A to A’ so that it is now OA’ = 2*OA, you should realise that OA = AA’ and AA’ has its tail to AB. So that’s the angle the between the tails that the question effectively wants.

I don't get it. It says between vectors OA and AB, but the angle between these two is OAB, the vector OA ends at A so why should it be the acute angle because the two and not the obtuse?

"Show that the lines with equations r = (5i - 2j + k) + t(2i - j - k) and r = (i + j - k) + s(-2 + i + j + 5k) are skew."

Is this an error in question, since the -2 is left on it's own and doesn't have an i, j or k?