Chapter 13.2, Problem 34E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

At what point do the curves **r**_{1}(*t*) = ⟨*t*, 1 − *t*, 3 + *t*^{2}⟩ and **r**_{2}(*s*) = ⟨3 − *s*, *s* − 2, *s*^{2}⟩ intersect? Find their angle of intersection correct to the nearest degree.

To determine

**To find:** The point of intersection of the curves

Explanation

Find the values of scalar parameters and substitute in any of the vector functions of curves to obtain the point of intersection of two curves.

As the two curves are intersected, equate *x*-, *y*-, and *z*-components of both the vectors and solve the expressions to obtain the values of scalar parameters

Equate *x*-component of vectors as follows.

Equate *y*-component of vectors as follows.

Equate *z*-component of vectors as follows.

Solve equation (2) and (3) as follows.

Rewrite the expression in equation (2) as follows.

Substitute

Solve the expression to obtain the value of scalar parameter

Substitute 1 for

The values of scalar parameters

**Calculation of point of intersection:**

As the *x*-, -, y-, and *z*-components of vector

Substitute 1 for

Thus, the point of intersection of the curves

**To find:** The angle of intersection of curves

**Solution:**

The angle of intersection of curves

The angle of intersection of two curves is the angle between the two tangent vectors to the curves.

As the point of intersection is obtained at

**Formula used:**

Write the expression to find the angle between two tangent vectors

Here,

**Calculation of tangent vector** **:**

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function

Use the following formulae to compute the expression.

Compute the expression

**Calculation of tangent vector** ** at the point of intersection:**

Substitute 1 for

**Calculation of tangent vector** **:**

Differentiate each component of the vector function

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