Find the component form of v given its magnitude and the angle it makes with the positive x-axis. Sketch v.
Magnitude: ||v||=3
angle: θ = 45°
as usual,
x = 3cosθ
y = 3sinθ
so, v = <3/√2 , 3/√2)
To find the component form of vector v, we can use the following formulas:
vₓ = ||v|| * cos(θ)
vᵧ = ||v|| * sin(θ)
Given that ||v|| = 3 and θ = 45°, we can substitute these values into the formulas to find the components of vector v:
vₓ = 3 * cos(45°)
vₓ = 3 * √(2)/2
vₓ = 3√2/2
vᵧ = 3 * sin(45°)
vᵧ = 3 * √(2)/2
vᵧ = 3√2/2
Therefore, the component form of vector v is (3√2/2, 3√2/2).
To sketch vector v, start by placing the initial point of the vector at the origin (0,0) on a coordinate plane. Then, from the origin, move 3√2/2 units to the right (positive x-direction) and 3√2/2 units up (positive y-direction). Mark this point as the terminal point of the vector. Finally, draw an arrow to indicate the direction of the vector.