Find a vector equation and parametric equations for the line segment that joins P to Q. P(0, - 1, 1), Q(1/2, 1/3, 1/4).

Solution:

Given, P(0, -1, 1), Q(1/2, 1/3, 1/4)

We have to find a vector equation and parametric equations for the line segment that joins P to Q.

Vector equation of a line segment joining the points with position vectors r₀ and r₁ is given by

r =(1 - t)r₀ + tr₁

Where, t ∈ [0, 1]

Here, r₀ = (0, -1, 1) and r₁ = (1/2, 1/3, 1/4)

So, r(t) = (1 - t)(0, -1, 1) + t(1/2, 1/3, 1/4)

r(t) = [((1 - t)(0) -1(1 - t) + 1(1 - t)] + [t/2, t/3, t/4]

r(t) = [0, (t - 1), (1 - t)] + [t/2, t/3, t/4]

On simplification,

r(t) = [(t/2 + 0), (t + t/3 - 1), (1 - t + t/4)]

r(t) = [t/2, (4t/3 - 1), (1 - 3t/4)]

Where, t ∈ [0,1]

The parametric equations for the line segment are

x = t/2, y = -1 + 4t/3, z = 1 - 3t/4

Where, t ∈ [0, 1]

Therefore, the vector and parametric equations are r(t) = [t/2, (4t/3 - 1), (1 - 3t/4)] and x = t/2, y = -1 + 4t/3, z = 1 - 3t/4.

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Find a vector equation and parametric equations for the line segment that joins P to Q. P(0, - 1, 1), Q(1/2, 1/3, 1/4).

Summary:

The vector equation and parametric equations for the line segment that joins P to Q. P(0, - 1, 1), Q(1/2, 1/3, 1/4) are r(t) = [t/2, (4t/3 - 1), (1 - 3t/4)] and x = t/2, y = -1 + 4t/3, z = 1 - 3t/4.