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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

Respuesta :

oladayoademosu

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:

https://brainly.com/question/25705666

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