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

Solution:

Let v be the vector of form <x, y>

Determinant |v| = 8, then

x² + y² = (8)²

x² + y² = 64 --- (1)

If it makes an angle π/3 with the positive x-axis, then the tangent relationship yields:

tan(π/3) = 1.732

y/x = 1.732

Squaring on both sides,

y²/x² = (1.732)²

y²/x² = 3

y² = 3x²

Put the value of y² in (1)

x² + 3x² = 64

4x² = 64

x² = 64/4

x² = 16

Taking square root,

x = √16

x = 4

As positive is given so neglecting negative value.

Now, y² = 3(4)²

= 3(16)

= 48

Taking square root,

y = √48

y = 4√3

Therefore, v can be represented in component form as v = <4, 4√3>.

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

Summary:

If v lies in the first quadrant and makes an angle pi/3 with the positive x-axis and |v| = 8, then v in component form is v = <4, 4√3>.