Vectors and conics
Vectors and conics

This free course is available to start right now. Review the full course description and key learning outcomes and create an account and enrol if you want a free statement of participation.

Free course

Vectors and conics

1.10 Further exercises

Example 12

Determine the equation of the line through each of the following pairs of points. Show that both equations can be written in the form ax + by = c, for some real numbers a, b and c, where a and b are not both zero.

  • (a) (−2, −4) and (1, 6).

  • (b) (0, 0) and (7, 3).

Answer

  • (a) Since (−2, −4) and (1, 6) lie on the line, its gradient is

  • It follows that the equation of the line is

  • which can be simplified to

  • that is,

  • 10x − 3y = −8.

  • This equation is of the desired form, with a = 10, b = −3 and c = −8. (Any multiple of these numbers is also a valid answer.)

  • (b) Since the line passes through the origin and the point (7, 3), it has an equation of the form y = mx, for some m. The coordinates of (7, 3) must satisfy the equation y = mx. Thus 3 = 7m, so

    .

  • Hence the equation of the line is

  • This can be written as

  • 3x − 7y − 0,

  • which is of the desired form, with a = 3, b = −7 and c = 0.

Example 13

Determine the values of k for which the lines 3x + 4y + 7 = 0 and 2x + ky = 3 are

  • (a) parallel,
  • (b) perpendicular.

Answer

The gradients of the lines 3 x + 4y + 7 = 0 and 2 x + ky = 3 are and , respectively.

Thus the lines are

  • (a) parallel if , that is, ;

  • (b) perpendicular if , that is, .

Example 14

Sketch the lines with the following equations, on a single diagram:

Answer

Example 15

Determine the coordinates of the points of intersection of the lines in Exercise 14.

Answer

Let A be the point of intersection of the lines y = −3x and . We equate the two expressions for y to obtain

Multiplying through by 3 gives

−9x = x + 6,

so

Since A lies on the line y = −3x, it follows that

So the point A has coordinates

Next, let B be the point of intersection of the lines and y − 3 = 3(x − 3). We rewrite the second equation, and equate the two expressions for y to obtain

Multiplying through by 3 and collecting terms gives

so

x = 3.

Since B lies on the line it follows that y = 3. So the point B has coordinates (3, 3).

Finally, let C be the point of intersection of the lines y = −3x and y − 3 = 3(x − 3). We rewrite the second equation, and equate the two expressions for y to obtain

−3x = 3(x − 3) + 3.

Collecting terms gives

6x = 6,

so

x = 1.

Since C lies on the line y = −3x, it follows that y = −3. So the point C has coordinates (1, −3).

Example 16

Find the distances between the vertices of the triangle formed by the points of intersection found in Exercise 15.

Answer

We use the Distance Formula given Section 1.5.

Since , B = (3, 3) and C = (1, −3),

Remark: In the triangle ABC,

AB2 + AC2 = BC2,

so BAC is a right angle.

Example 17

Determine whether each of the pairs of planes given by the following equations intersect, are parallel, or coincide.

  • (a) x = 1 and y = 2.

  • (b) z = 1 and z = 3.

Illustrate your answer to each part with a sketch.

Answer

  • (a) The planes with equations x = 1 and y = 2 are parallel to the (y, z)-plane and the (x, z)-plane, respectively. They intersect in a line, as shown.

  •  

  • (b) The planes with equations z = 1 and z = 3 are both parallel to the (x, y)-plane. They are parallel to each other, as shown.

Example 18

Determine the distance between the points (1, −2, 3) and (−2, 3, −1) in 3.

Answer

We use the Distance Formula given in Section 1.9. The required distance is thus

M208_1

Take your learning further

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses.

If you are new to university level study, find out more about the types of qualifications we offer, including our entry level Access courses and Certificates.

Not ready for University study then browse over 900 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus