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
<h3>a. How to resolve vector v into components?</h3>
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,
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
<h3>b. How to resolve vector w into components?</h3>
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
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