Vectors

Question 1

(a,b,c).(x,y,z)?

Answer 1

= ax + by + cz

Question 2

Angle between two vectors?

Answer 2

Cos theta = (a.b)/lallbl

Question 3

What is the dot product of two perpendicular lines

Answer 3

The dot product is 0.

Cos 90 = 0 = (a.b)/lallbl

Question 4

Given a = (2, 5, -4) and b = (4, -8, 5) find the vector perpendicular to a and b

Answer 4

Let this vector be (x, y, z)
Then (2, 5, -4).(x, y ,z) = 0
And (4, -8, 5).(x, y, z) = 0
Let z = 1
2x + 5y = 4
4x - 8y = -5
=> x = 7/4, y = 3/2, z = 1
So a possible vector is (7/4, 3/2, 1)

Question 5

Equation of plane in vector form?

Answer 5

Let the normal to the plane = n
a and r are points on the plane
AR . n = 0
=> (r - a).n = 0
r.n - a.n = 0
r.n = a.n (often denoted by d)

Question 6

Equation of a plane in Cartesian form?

Answer 6

n = (n1,n2,n3) and a = (a1,a2,a3) and r = (x,y,z) (any vector on the plane)
Then for r.n - a.n = 0:
n1x + n2y + n3z -a.n = 0

n1x + n2y + n3z + d = 0 (where d = -a.n)

Question 7

A = (2, 3, -5) on a plane, normal to the plane is (-4,2,1) find Cartesian equation

Answer 7

n1x + n2y + n3z + d = 0
Cartesian eqn = -4x + 2y + z + d = 0
To find d, either sub in a point on the plane:
-4(2) + 2(3) + 1(-5) + d = 0 => d = 7
Or d = -a.n = -(2,3,5).n = 7

Question 8

What happens if you change d in the Cartesian equation

What do the coefficients of x,y,z and d determine in the Cartesian equation

Answer 8

The planes are parallel (but still have the same normal)

X,y,z determine the direction of the plane, d determines its location

Question 9

What is the equation of the plane parallel to 3x - y + 2z + 5 = 0 which contains the point (1,0,-2)? The normal is (3,-1,2)

Answer 9

Any parallel plane has equation 3x - y + 2z + d = 0
d = -a.n
= - (3x1 + -1x0 + 2x-2)
d = 1
Equation is 3x - y + 2z + 1 = 0

Question 10

Given 3 points A, B and C on a plane, how would you find the equation?

Answer 10

Substitute the points into ax + by + cz + d = 0

Question 11

A plane contains the points (1,1,1), (1,-1,0) and (-1,0,2). Find the equation of the plane

Answer 11

sub each point into Cartesian equation:
a(1) + b(1) + c(1) = -d
=> a + b + c = -d
a - b = -d
-a + 2c =-d
See photograph

Question 12

How do you find the angle between two planes?

Answer 12

Angle between planes is equal to the angles between their normals. Find their normals and use the cos theta thing

Question 13

What are the 5 ways that three planes can intersect?

Answer 13

• if two of the planes are parallel, there are 2 possibilities for the arrangement of the third:
  It can either be parallel to the other two (so all three parallel) or it can cut the other two (two parallel, and one cuts both - there is no common intersection between all three therefore no solutions)
• if none of the planes are parallel, there are three possibilities:
  The planes intersect in a single point (so only one distinct solution)
  The planes form a sheaf (the three planes share a common line, so there are an infinite number of points of intersection)
  The planes form a triangular prism (each pair of planes intersects in a straight line; the three lines are parallel. So there is no common intersection)

Question 14

Find the unique point of intersection of the three planes:
2x+y-2z=5, x+y+z=1, x-y-3z=-2

Answer 14

Set up matrix simultaneous equations
Rearrange to find x, y and z. This the the unique point of intersection

Question 15

If the matrix of coefficients for the three points (when finding their intersection) is non-singular…

Answer 15

The planes intersect at a single point

Question 16

If the matrix of coefficients for the three points (when finding their intersection) IS singular…

Answer 16

The plane must be arranged in one of the other four possible arrangements:
Are there infinite solution when you type the sim. equation into calc?
- yes: sheaf
-no: are all three rows of the coefficient matrix multiples of each other?
- yes: 3 parallel planes
- no: are two rows multiples of each other?
- yes: 2 parallel planes
- no: triangular prism

Question 17

Show that the planes do not have a unique intersection point:

Therefore determine their arrangement:

Answer 17

Find det(M) of the coefficient matrix. If this is singular (=0), it means there is no unique point

Check for multiples. No multiples?
Check for infinite solutions using sim eq solver on calc. No solutions?
Then it must be a triangular prism