Question

find a vector equation and parametric equations for the line segment that joins p to q. p(a, b, c), q(u, v, w) vector equation r(t) = parametric equations (x(t), y(t), z(t)) =

Answer 1

The vector equation for the line segment is r(t) = (a, b, c) + t((u - a), (v - b), (w - c)). The parametric equations are x(t) = a + t(u - a), y(t) = b + t(v - b), z(t) = c + t(w - c).

Step-by-step explanation:

To find the vector equation and parametric equations for the line segment that joins point P(a, b, c) to point Q(u, v, w), we can use the formula:

r(t) = P + t(Q - P)

where P and Q are the position vectors of points P and Q, and t is a parameter.

Using this formula, the vector equation for the line segment is:

r(t) = (a, b, c) + t((u - a), (v - b), (w - c))

The parametric equations are:

x(t) = a + t(u - a),

y(t) = b + t(v - b),

z(t) = c + t(w - c)

Answer 2

The vector equation for the line segment joining P to Q is:r(t) = (a + t(u - a), b + t(v - b), c + t(w - c))

The parametric equations for the line segment joining P to Q are:

x(t) = a + t(u - a)

y(t) = b + t(v - b)

z(t) = c + t(w - c)

How to find the vector equation and parametric equations

To find the vector equation and parametric equations for the line segment that joins point P(a, b, c) to point Q(u, v, w), use the parameter t to represent the position along the line segment.

Vector equation:

The vector equation for a line segment can be written as:

r(t) = P + t(Q - P)

In this case, P = (a, b, c) and Q = (u, v, w).

Substitute these values into the equation

r(t) = (a, b, c) + t((u, v, w) - (a, b, c))

Simplifying, we have:

r(t) = (a, b, c) + t(u - a, v - b, w - c)

Therefore, the vector equation for the line segment joining P to Q is:

r(t) = (a + t(u - a), b + t(v - b), c + t(w - c))

Parametric equations:

To obtain the parametric equations, separate the vector equation into three equations representing the x, y, and z components.

x(t) = a + t(u - a)

y(t) = b + t(v - b)

z(t) = c + t(w - c)

Therefore, the parametric equations for the line segment joining P to Q are:

x(t) = a + t(u - a)

y(t) = b + t(v - b)

z(t) = c + t(w - c)

These equations allow us to find the coordinates of any point on the line segment as we vary the parameter t from 0 to 1.