Question

Resolve the following vectors into components:

(a) The vector v in 2-space of length 3 pointing up at an angle of π/4 measured from the positive x-axis.

v = _ i + _ j

(b) The vector w in 3-space of length 1 lying in the yz-plane pointing upward at an angle of 2π/3 measured from the positive y-axis.

v = _ i + _ j + _ k

1. a. Vector v resolved into components is v =  (3√2/2)i +  (3√2/2)j
b. Vector w resolved into components is w = 0i + (1/2)j +  (√3/2)k

### a. How to resolve vector v into components?

A vector in 2 dimension is given by r = xi + yj
where
• x = x- component = rcosθ and
• y = y-component = rsinθ
• r = length of vector and
• θ = angle between vector and x – axis.
Given that the vector v in 2-space of length 3 pointing up at an angle of π/4 measured from the positive x-axis, we have that,
• r = 3 and
• θ = π/4
So, v = xi + yj
x =  rcosθ = 3cosπ/4
= 3 × 1/√2
= 3/√2 × √2/√2
= 3√2/2
y =  rsinθ
= 3sinπ/4
= 3 × 1/√2
= 3/√2 × √2/√2
= 3√2/2
So, v =  (3√2/2)i +  (3√2/2)j
Vector v resolved into components is v =  (3√2/2)i +  (3√2/2)j

### b. How to resolve vector w into components?

A vector in 3 dimension is given by r = xi + yj + zk
where
• x = x- component = rsinαcosθ and
• y = y-component = rsinαsinθ
• z = z-component = rcosα
• r = length of vector and
• θ = angle between vector and x – axis.
• α = angle between vector and z – axis
Given that the vector w in 3-space of length 1 lying in the yz-plane pointing upwards at an angle of 2π/3 measured from the positive x-axis, we have that,
• r = 1 and
• θ = π/2  (since the vector is in the yz-plane)
• Now, π – 2π/3 = π/3(angle between w and negative y-axis)
• so, α = π/2 – π/3 = π/6(angle between w and positive z-axis)
So, v = xi + yj + zk
x =  rsinαcosθ
= 1 × sin(π/6)cos(π/2)
= 1 × 1/2 × 0
= 0
y = rsinαcosθ
= 1 × sin(π/6)sin(π/2)
= 1 × 1/2 × 1
= 1/2
z = rcosα
= 1 × cos(π/6)
= 1 × √3/2
= √3/2
So, w =  0i + (1/2)j +  (√3/2)k
Vector w resolved into components is w = 0i + (1/2)j +  (√3/2)k