Chapter 11: Q.53 (page 872)

Find all points of intersection between the vector function
$r (t) = t, t 2, t 3$ and the plane defined by $3 x - 3 y + z = 1 .$

Short Answer

Expert verified

$3 x - 3 y + z = 1$

$(1, 1, 1), (1, 1, 1) and (1, 1, 1)$

Step by step solution

Given Information

The objective of this problem is to find all point of intersection between the vector function and the plane.

Vector function

$r (t) = (t, t^{2}, t^{3}) The plane is 3 x - 3 y + z = 1$

Calculation

Substitute $x = t, y = t^{2} a n d z = t^{3}$ in the equation of plane

$3 t - 3 t^{2} + t^{3} = 1 t^{3} - 1 - 3 t^{2} + 3 t = 0 (t - 1) (t^{2} + t + 1) - 3 t (t - 1) = 0 (t - 1) (t^{2} - 2 t + 1) = 0 {(t - 1)}^{3} = 0 t = 1, 1, 1$

Thus the points of intersection are

localid="1650021665961" $(1, 1, 1), (1, 1, 1) and (1, 1, 1)$

This shows that the vector function passes through the plane at a single point three times.

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Find all points of intersection between the vector function
$r (t) = t, t 2, t 3$ and the plane defined by $3 x - 3 y + z = 1 .$

Short Answer

Step by step solution

Given Information

Calculation

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Find all points of intersection between the vector functionr(t)=t,t2,t3and the plane defined by 3x−3y+z=1.

Short Answer

Step by step solution

Given Information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Decision Maths

Logic and Functions

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find all points of intersection between the vector function
$r (t) = t, t 2, t 3$ and the plane defined by $3 x - 3 y + z = 1 .$