If v lies in the first quadrant and makes an angle π/3 with the positive x-axis and |v| = 8, find v in component form.

To find the vector v in component form, we can use the relationship between the magnitude of the vector, the angle it makes with the positive x-axis, and its components.

Given that the magnitude of the vector v is 8 and it makes an angle of π/3 radians with the positive x-axis, we can use the following formulas:

• vx = |v| * cos(θ) (for the x-component)
• vy = |v| * sin(θ) (for the y-component)

Substituting in the values:

• vx = 8 * cos(π/3)
• vy = 8 * sin(π/3)

Now, we calculate the cosine and sine values:

• cos(π/3) = 1/2
• sin(π/3) = √3/2

Plugging these values back into our equations gives us:

• vx = 8 * (1/2) = 4
• vy = 8 * (√3/2) = 4√3

Therefore, the vector v in component form is:

v = (4, 4√3)

In summary, the components of the vector v are 4 in the x-direction and 4√3 in the y-direction, placing it firmly in the first quadrant.