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In this lesson, we will learn how to find the equation of a straight line in parametric form using a point on the line and the vector direction of the line.

Q1:

Find the parametric equations of the straight line that passes through the point ( β 9 , 8 ) with direction vector ( 4 , β 7 ) .

Q2:

Find the parametric equations of the straight line that makes an angle of 1 3 5 β with the positive π₯ -axis and passes through the point ( 1 , β 1 5 ) .

Q3:

Consider the line shown that passes through the point ( 3 , 4 ) and makes an angle of 45 degrees with the positive π₯ -axis.

Suppose that the distance between ( 3 , 4 ) and any point ( π₯ , π¦ ) on the line is π .

Write, in terms of π , an expression for the horizontal distance π₯ β 3 between the two points.

Write, in terms of π , an expression for the vertical distance π¦ β 4 between the two points.

Hence, write a pair of parametric equations which describe the line.

Find the coordinates of the point on the line which is at a distance of 4 from ( 3 , 4 ) .

Q4:

Write a pair of parametric equations with parameter π describing the shown line.

Q5:

Write the parametric equation of the straight line that passes through the point ( π , π ) and makes an angle of π with the positive π₯ -axis as shown.

Q6:

The equations π₯ = 2 π‘ + 1 , π¦ = β 3 π‘ + 2 parameterize the line segment between ( 1 , 2 ) and ( 3 , β 1 ) over the interval 0 β€ π‘ β€ 1 .

Which of the following is a parameterization of the line segment on β 1 β€ π‘ β€ 0 ?

Which of the following is a parameterization of the line segment on 0 β€ π‘ β€ 1 that starts at ( 3 , β 1 ) and ends at ( 1 , 2 ) ?

Which of the following is a parameterization of the line segment on 0 β€ π‘ β€ 2 ?

If the parameterizations you have given above correspond to a particle moving along the line segment, how does the parameterization over interval 0 β€ π‘ β€ 2 relate to the one over 0 β€ π‘ β€ 1 ?

Q7:

Consider the points π΄ , π΅ , and πΆ and the line segments in the figure.

Give the parameterization of π΄ π΅ over the interval 1 β€ π‘ β€ 3 .

Give the parameterization of π΅ πΆ over the interval 3 β€ π‘ β€ 5 .

Find functions π and π defined for 1 β€ π‘ β€ 5 so that π₯ = π ( π‘ ) , π¦ = π ( π‘ ) parameterizes the given path from π΄ to πΆ .

Q8:

Let π΄ = ( 1 , 1 ) and π΅ = ( 1 , 3 ) . Which of the following is a parameterization of π΄ π΅ over 0 β€ π‘ β€ 1 that starts at π΄ and ends at π΅ .

Q9:

Let π΄ = ( 1 , 1 ) and π΅ = ( 1 , 2 ) . Which of the following is a parameterization of π΄ π΅ over 0 β€ π‘ β€ 1 that starts at π΅ and ends at π΄ .

Q10:

Find the parameterization π₯ = π ( π‘ ) , π¦ = π ( π‘ ) of the path π΄ , π΅ , πΆ , π· using the interval 1 β€ π‘ β€ 9 .

Q11:

Let π΄ = ( 1 , 1 ) and π΅ = ( 1 , 2 ) . Find the parameterization of π΄ π΅ over 0 β€ π‘ β€ 1 that starts at π΄ and ends at π΅ .

Q12:

A cube with side 3 sits with a vertex at the origin and three sides along the positive axes. Find the parametric equations of the main diagonal from the origin.

Q13:

True or False: There is only one way to parameterize the line segment from ( 1 , 2 ) to ( 3 , β 1 ) .

Q14:

Find the parametric equations of the straight line that passes through the point ( 9 , β 7 ) with direction vector ( 3 , 2 ) .

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