Further Vectors

Question 1

How do you calculate the intersection of a line and a plane?

Answer 1

Calculate lambda by substituting the values of ‘x’, ‘y’, and ‘z’ from the line equation into the equation of the plane

Question 2

How do you calculate the angle between a line and a plane?

Answer 2

Find the angle between the direction vector of the line and the normal vector of the plane using the dot product
Take this away from 90^o

Question 3

What is the formula for the vector product of a x b?

Answer 3

a₂b₃ - a₃b₂
a₃b₁ - a₁b₃
a₁b₂ - a₂b₁

Question 4

What does the vector product equal?

Answer 4

a x b = IaI * IbI * sin(x) * n
‘x’ is the angle between ‘a’ and ‘b’
‘n’ is the unit vector perpendicular to ‘a’ and ‘b’

Question 5

What is the vector product?

Answer 5

The product of ‘a’ and ‘b’ is perpendicular to both ‘a’ and ‘b’

Question 6

How do you calculate the distance between a point and a line in two dimensions?

Answer 6

Point : (x₁, y₁)
Line : ax + by + c = 0

Distance = I ax₁ + by₁ + c I / Sqrt( a² + b²)

Question 7

How do you calculate the distance between a point and a plane?

Answer 7

Point : (x₁, y₁, z₁)
Plane : n ₁x + n₂y + n₃z + d = 0

Distance = I n₁x₁ + n₂y₂ + n₃z₃ + d I / Sqrt( n₁² + n₂² + n₃²)

Question 8

How do you calculate the distance between skew lines?

Answer 8

d₁ is parallel to the first line
a₁ is a position vector of a point on the first line

I d₁ x d₂ / I d₁ x d₂ I . (a₁ - a₂) I

Question 9

What is the formula for the shortest distance of a point to a line?

Answer 9

‘AP’ is the vector of a point ‘A’ on the line to point ‘P’
‘d’ is the direction vector of the line

D = I AP x d I / IdI