   Chapter 12.3, Problem 30E

Chapter
Section
Textbook Problem

# Find the acute angle between the lines.30. x + 2y = 7, 5x − y = 2

To determine

To find: The acute angle between lines.

Explanation

Given:

x+2y=7 (1)

5xy=2 (2)

Formula:

Write the expression for line equation.

y=mx+c (3)

Here,

c is constant, and

m is slope.

Write the expression to find ab in terms of θ .

ab=|a||b|cosθ

Here,

|a| is the magnitude of a vector,

|b| is the magnitude of b vector, and

θ is the angle between vectors a and b.

Rearrange equation.

θ=cos1(ab|a||b|) (4)

Consider a general expression to find dot product between two two-dimensional vectors.

ab=a1,a2b1,b2

ab=a1b1+a2b2 (5)

Consider a general expression to find magnitude of a two dimensional vector that is a=a1,a2 .

|a|=a12+a22 (6)

Similarly, Consider a general expression to find magnitude of a two dimensional vector that is b=b1,b2 .

|b|=b12+b22 (7)

The acute angle between the lines=180°θ (8)

Rearrange equation (1).

y=12x+72

Compare y=12x+72 with equation (3).

m=12

The vector parallel to the line is a=2,1 .

Rearrange equation (2).

y=5x2

Compare y=5x2 with equation (3)

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