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1. A line passes through points A(1,2,4) and B(2,3,6). a. Determine a vector equation for this line. b. Determine the respective parametric equations of this line. c. Determine a vector equation of a of the line in parametric form. Also, write the equation in non - parametric form.

User Leonora
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Answer:

Explanation:

a. To determine a vector equation for the line passing through points A(1,2,4) and B(2,3,6), we can find the direction vector of the line by subtracting the coordinates of the two points.

Direction vector:

d = B - A = (2, 3, 6) - (1, 2, 4) = (1, 1, 2)

Now, we can express the vector equation for the line as:

r = A + td

where r is a position vector on the line, t is a parameter, A is a point on the line (A(1,2,4)), and d is the direction vector we found.

The vector equation for the line is: r = (1,2,4) + t(1,1,2)

b. To determine the respective parametric equations of the line, we can assign variables to each coordinate of the point A and the direction vector.

Let x = 1 + t, y = 2 + t, and z = 4 + 2t.

The respective parametric equations of the line are:

x = 1 + t

y = 2 + t

z = 4 + 2t

c. The vector equation of the line in parametric form is r = (1,2,4) + t(1,1,2).

To write the equation in non-parametric form, we can express x, y, and z in terms of t:

x = 1 + t

y = 2 + t

z = 4 + 2t

Rearranging the equations, we can eliminate t:

t = x - 1

t = y - 2

t = (z - 4)/2

Equating the expressions for t, we have:

x - 1 = y - 2 = (z - 4)/2

This is the non-parametric equation of the line.

In summary:

a. Vector equation for the line: r = (1,2,4) + t(1,1,2)

b. Parametric equations of the line: x = 1 + t, y = 2 + t, z = 4 + 2t

c. Vector equation of the line in parametric form: r = (1,2,4) + t(1,1,2)

Non-parametric equation of the line: x - 1 = y - 2 = (z - 4)/2

User Corey Farwell
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