Resolve the following vectors into components: (a) The vector v in 2-space of length 3 pointing up at an angle of

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

in progress 0
Orla Orla 1 month 2022-12-28T23:25:58+00:00 1 Answer 0 views 0

Answer ( 1 )

    0
    2022-12-28T23:27:27+00:00
    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:
    #SPJ1

Leave an answer

Browse

Giải phương trình 1 ẩn: x + 2 - 2(x + 1) = -x . Hỏi x = ? ( )