Answers

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

   Learn more about vectors here:

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