Vector

Key Concepts

• Understand vector, initial and terminal points
• Identify vector components
• Use a vector to translate a figure.
• Solve a multi-step problem.

Introduction

Vector

Another way to describe a translation is by using a vector. A vector is a quantity that has both direction and magnitude, or size. A vector is represented in the coordinate plane by an arrow drawn from one point to another.

The diagram shows a vector named AB, read as “vector AB.”

The initial point, or starting point, of the vector is A.

The terminal point, or ending point, of the vector is B.

Example 1:

Name the vector, and initial and final point.

Solution:

The diagram shows a vector named DC, read as “vector DC.”

The initial point, or starting point, of the vector is D.

The terminal point, or ending point, of the vector is C.

Example 2:

Name the vector, and initial and final point.

Solution:

The diagram shows a vector named CD, read as “vector CD.”

The initial point, or starting point, of the vector is C.

The terminal point, or ending point, of the vector is D.

Example 3:

Name the vector, and initial and final point.

Solution:

The diagram shows a vector named PQ, read as “vector PQ.”

The initial point, or starting point, of the vector is P.

The terminal point, or ending point, of the vector is Q.

Vector components

Let us understand this concept with the help of a graph:

The diagram shows a vector named AB , read as “vector AB.”

• The initial point, or starting point, of the vector is A.
• The terminal point, or ending point, of the vector is B.

The horizontal distance of the vector AB is known as the horizontal component.

The vertical distance of the vector AB is known as the vertical component.

• On the x-axis, sign convention,

If the vector moves right, then + component

If the vector moves left, then – component

• On the Y-axis, sign convention,

If the vector moves upward, then + component

If the vector moves down, then – component.

• The component form of a vector combines the horizontal (Moves right from the initial point) and vertical components (Moves up from the initial point).

So, the component form of AB is 11,10

Example 1:

Name the vector and write its component form.

Solution:

The diagram shows a vector named BC, read as “vector BC.”

The initial point, or starting point, of the vector is B.

The terminal point, or ending point, of the vector is C.

Horizontal component = +9 units (It moves 9 units right)

Vertical component = –2 units (It moves 2 units downwards)

So, the component form is 〈9, –2〉.

Example 2:

Name the vector and write its component form.

Solution:

The diagram shows a vector named ST, read as “vector ST.”

The initial point, or starting point, of the vector is S.

The terminal point, or ending point, of the vector is T.

Horizontal component = –8 units (It moves 8 units left)

Vertical component = 0

So, the component form is〈–8, 0〉.

Translate a figure using vectors:

To translate any figure using vector form, firstly, find out the component form.

With the help of the component form, translate the figure horizontally and vertically.

Now, label the image formed.

Now, we can notice that the vectors drawn from preimage to image vertices are parallel.

Example 1:

The vertices of ΔABC are A(0, 3), B(2, 4), and C(1, 0). Translate ΔABC using the vector 〈5, –1〉.

Solution:

First, graph ΔABC. Use 〈5, –1〉 to move each vertex 5 units to the right and 1 unit down.

Label the image vertices.

Draw ΔA’B’C’.

Notice that the vectors drawn from preimage to image vertices are parallel.

Solve a multi-step problem:

Example:

A boat heads out from point A on one island toward point D on another. The boat encounters a storm at B, 12 miles east and 4 miles north of its starting point. The storm pushes the boat off course to point C, as shown.

1. Write the component form of AB.
2. Write the component form of BC.
3. Write the component form of the vector that describes the straight-line path from the boat’s current position C to its intended destination D.

Solution:

1. The component form of the vector from A(0, 0) to B(12, 4) is

AB = 5〈12 – 0, 4 – 0〉 = 〈12, 4〉.

• The component form of the vector from B(12, 4) to C(16, 2) is

BC= 〈16 – 12, 2 – 4〉 = 〈4, –2〉.

• The boat is currently at point C and needs to travel to D. The component form of the vector from C(16, 2) to D(18, 5) is

CD = 〈18 – 16, 5 – 2〉 = 〈2, 3〉.

Examples:

1. Name the vector and write its component form.

Solution:

The diagram shows a vector named CD, read as “vector CD”.

The initial point, or starting point, of the vector is C.

The terminal point, or ending point, of the vector is D.

Horizontal component = 7 units (It moves 7 units right)

Vertical component = –3 (It moves 3 units down)

So, the component form is 〈7, –3〉.

• Name the vector and write its component form.

Solution:

The diagram shows a vector named RT, read as “vector RT”.

The initial point, or starting point, of the vector is R.

The terminal point, or ending point, of the vector is T.

Horizontal component = –2 units (It moves 2 units left)

Vertical component = –4 (It moves 4 units down)

So, the component form is 〈–2, –4〉.

• The vertices of Δ PST are P(2, 2), S(5, 3), and T(9, 1). Translate Δ PST using the vector 〈–2, 6〉.

Solution:

First, graph ΔPST. Use 〈–2, 6〉 to move each vertex 2 units to the left and 6 units up.

Label the image vertices.

Draw ΔP’S’T.

Notice that the vectors drawn from preimage to image vertices are parallel.

Summary

• Vector: Another way to describe a translation is by using a vector. A vector is a quantity that has both direction and magnitude, or size. A vector is represented in the coordinate plane by an arrow drawn from one point to another.
• The starting point of the vector is termed as the initial point.
• The ending point of the vector is termed as the terminal point.
• On the x-axis, sign convention,

If the vector moves right, then + component

If the vector moves left, then – component

• On the Y-axis, sign convention,

If the vector moves upward, then + component

If the vector moves down, then – component.

• The component form of a vector combines the horizontal (Moves right from the initial point) and vertical components (Moves up from the initial point).

Exercise

• Name the vector and write its component form.

• Name the vector and write its component form.

• Use the point P(–3, 6). Find the component form of the vector that describes the translation to P’ (–4, 8).
• Use the point P(–3, 6). Find the component form of the vector that describes the translation to P’ (–3, –5).
• The vertices of PQR are P(2, 5), Q(6, 3), and R(4, 0). Translate PQR using the given vector. Graph PQR and its image. Component form = 〈–3, –4〉.
• The vertices of PQR are P(2, 5), Q(6, 3), and R(4, 0). Translate PQR using the given vector. Graph PQR and its image. Component form = 〈–2, –4〉.
• The vertices of a rectangle are Q(2, 23), R(2, 4), S(5, 4), and T(5, 23). Translate QRST
  3 units left and 2 units down.
• The vertices of a rectangle are Q(2, 23), R(2, 4), S(5, 4), and T(5, 23). Translate QRST
  3 units left and 2 units down. Find the areas of QRST and Q’R’S’T’.
• You are snowshoeing in the mountains. The distances in the diagram are in miles. Write the component form of the vector from the cabin to the ski lodge.

Concept Map

What we have learned

• Understand vector, initial and terminal points
• Identify vector components
• Use a vector to translate a figure.
• Solve a multi-step problem