vectors question please help

The tent is modelled as a triangular prism ABCDEF shown in Figure 3. The side ABCD is modelled by part of the plane with Cartesian equation 2x + 3y – 4z = 1 The side ABFE is modelled by part of the plane with Cartesian equation 8x + 12y + 15z = 252 (a) Find, according to the model, the acute angle between these two sides of the tent. Give your answer to the nearest degree. (3) These two sides of the tent meet along the straight line AB. (b) Show, according to the model, that the point P (6, 7, 8) lies on this straight line. (2) One end of a rope is attached to the top of the tent at the point P. The other end is pegged into the ground at the point Q. The rope is modelled as a straight line and, according to the model, Q has coordinates (–4, –3, 0) (c) Find, according to the model, the acute angle between the rope PQ and the side ABCD of the tent. Give your answer to the nearest degree. (5)

Reply 1

username4848156

OP

9

This is the image of the planes

Screenshot 2020-04-28 at 16.58.23.png90.4KB

OP

Original post by Dashydundah

The tent is modelled as a triangular prism ABCDEF shown in Figure 3. The side ABCD is modelled by part of the plane with Cartesian equation 2x + 3y – 4z = 1 The side ABFE is modelled by part of the plane with Cartesian equation 8x + 12y + 15z = 252 (a) Find, according to the model, the acute angle between these two sides of the tent. Give your answer to the nearest degree. (3) These two sides of the tent meet along the straight line AB. (b) Show, according to the model, that the point P (6, 7, 8) lies on this straight line. (2) One end of a rope is attached to the top of the tent at the point P. The other end is pegged into the ground at the point Q. The rope is modelled as a straight line and, according to the model, Q has coordinates (–4, –3, 0) (c) Find, according to the model, the acute angle between the rope PQ and the side ABCD of the tent. Give your answer to the nearest degree. (5)

Have you done dot product to find the angle between the normals to the planes?

Reply 4

username4848156

OP

9

Original post by mqb2766

Have you done dot product to find the angle between the normals to the planes?

yeh I've got an angle of 86 for part a)
I'm struggling with part c
and for part b do I just have to show that point P satisfies the two planes?

Reply 5

mqb2766

21

Original post by Dashydundah

yeh I've got an angle of 86 for part a)
I'm struggling with part c
and for part b do I just have to show that point P satisfies the two planes?

b) Its a bit hard to read but P will satisy both plane equations if it lies on both.
C) For P, find its base (z=0), and that together with P and Q will define a vertical right angle triangle so ... or
If you want the angle with the tent side, dot it with the normal to abcd.

(edited 3 years ago)

Reply 6

username4848156

OP

9

C) One end of a rope is attached to the top of the tent at the point P. The other end is pegged into the ground at the point Q.
The rope is modelled as a straight line and, according to the model, Q has coordinates (–4, –3, 0)
Find, according to the model, the acute angle between the rope PQ and the side ABCD of the tent. Give your answer to the nearest degree. (5)

Reply 7

username4848156

OP

9

so for b) do I just sub P (6, 7, 8) into the equations of the two planes and show that an identity exists?
Or do I need to find a straight line of intersection between the two planes

Reply 8

mqb2766

21

Original post by Dashydundah

so for b) do I just sub P (6, 7, 8) into the equations of the two planes and show that an identity exists?
Or do I need to find a straight line of intersection between the two planes

Sub the x,y,z values if P into the left hand side of the plane equation and show it equals the right hand side value

Reply 9

username4848156

OP

9

I don't understand what you mean for part c
please could you show me

Reply 10

mqb2766

21

Original post by Dashydundah

I don't understand what you mean for part c
please could you show m

Eithet
* Set the z component of P to 0. That point, R, is the poles base and PQR is a right angled triangle which gives the string angle with the ground. Use that and the tent side angle to get their difference.
* Or the line PQ can be dotted with the normal to the plane to directly find the angle between them.
Have a go at either/both?