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Unit notes and practice questions

Grade 9 Mathematics unit 8: Vectors in Two Dimensions

Everything here is for unit 8 of the Grade 9 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.

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4
Cards
10
Practice questions on unit 8
247–278
The textbook pages for unit 8

What unit 8 covers

  1. 1.Scalar and Vector Quantities
  2. 2.Representation of a Vector
  3. 3.Vectors Operations
  4. 4.Position Vector
  5. 5.Applications of Vectors in Two Dimensions

Short notes for unit 8

The cards filed under this chapter on the Grade 9 Mathematics page. Open one at a time.

When you use it

When combining two non parallel vectors to find their resultant.

Watch out

Never add the magnitudes of two vectors directly using scalar addition unless they act in the exact same direction.

Drafted from Grade 9 Mathematics, pages 247-262, then checked twice before it went up

∣a∣=x2+y2|\mathbf{a}| = \sqrt{x^2 + y^2}
∣a∣|\mathbf{a}|
magnitude of vector aunits\text{units}
xx
horizontal component of vector aunits\text{units}
yy
vertical component of vector aunits\text{units}

When you use it

Use to find the length or magnitude of a two dimensional vector given in column form.

Watch out

Always square negative components to get a positive value before adding under the square root.

Drafted from Grade 9 Mathematics, pages 247-262, then checked twice before it went up

∣PQ⃗∣=(x2−x1)2+(y2−y1)2|\vec{PQ}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
∣PQ⃗∣|\vec{PQ}|
magnitude of vector PQunits\text{units}
x1x_1
x coordinate of initial point Punits\text{units}
y1y_1
y coordinate of initial point Punits\text{units}
x2x_2
x coordinate of terminal point Qunits\text{units}
y2y_2
y coordinate of terminal point Qunits\text{units}

When you use it

Use to calculate the magnitude of a vector between given initial and terminal points on a coordinate plane.

Watch out

Subtract the initial point coordinates from the terminal point coordinates, not the other way around, to preserve direction.

Drafted from Grade 9 Mathematics, pages 247-262, then checked twice before it went up

AB⃗+BC⃗=AC⃗\vec{AB} + \vec{BC} = \vec{AC}
AB⃗\vec{AB}
first vector from initial point A to terminal point Bunits\text{units}
BC⃗\vec{BC}
second vector from initial point B to terminal point Cunits\text{units}
AC⃗\vec{AC}
resultant vector connecting initial point A to terminal point Cunits\text{units}

When you use it

Use to find the resultant vector when the initial point of the second vector connects to the terminal point of the first vector.

Watch out

Ensure the tail of the second vector is placed at the head of the first vector before connecting initial to terminal.

Drafted from Grade 9 Mathematics, pages 247-262, then checked twice before it went up

Practice questions on unit 8

10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.

Question 1

Which of the following describes a scalar quantity?

  1. A.It is fully described by magnitude alone.
  2. B.It is fully described by direction alone.
  3. C.It must have both magnitude and direction.
  4. D.It has direction but no numerical value.
Show the answer and why

Answer: A It is fully described by magnitude alone.

Why

A scalar is a quantity that is fully described by a magnitude (or numerical value) alone, without any direction.

Question 2

Which of the following physical quantities is a vector quantity?

  1. A.Mass of a stone
  2. B.Temperature of a room
  3. C.Velocity of a boat
  4. D.Density of pure water
Show the answer and why

Answer: C Velocity of a boat

Why

Velocity requires both a magnitude and a direction, so it is a vector quantity. Mass, temperature, and density are scalar quantities.

Question 3

When are two vectors said to be equal?

  1. A.When they have the same magnitude and the same direction.
  2. B.When they have the same magnitude but opposite directions.
  3. C.When they have different magnitudes but the same line.
  4. D.When their initial points are at the origin.
Show the answer and why

Answer: A When they have the same magnitude and the same direction.

Why

Two vectors are equal if and only if they have the same magnitude and the same direction.

Question 4

What is the opposite vector of a vector a?

  1. A.A vector with half the magnitude of a in the same direction.
  2. B.A vector with the same magnitude as a, but opposite in direction.
  3. C.A vector with double the magnitude of a in any direction.
  4. D.A vector with zero magnitude.
Show the answer and why

Answer: B A vector with the same magnitude as a, but opposite in direction.

Why

The opposite of a vector has the same magnitude as the original vector, but it points in the exact opposite direction. It is denoted by -a.

Question 5

What is the magnitude of the column vector u = (4, 3)T, where 4 is the horizontal component and 3 is the vertical component?

  1. A.7 units
  2. B.5 units
  3. C.25 units
  4. D.1 unit
Show the answer and why

Answer: B 5 units

Why

The magnitude of a column vector (x, y) is |u| = √(x2 + y2). Here, |u| = √(42 + 32) = √(16 + 9) = √25 = 5 units.

Question 6

Vectors that have the same direction or opposite directions are called:

  1. A.Perpendicular vectors
  2. B.Parallel vectors
  3. C.Equal vectors
  4. D.Unit vectors
Show the answer and why

Answer: B Parallel vectors

Why

By definition, vectors that have the same direction or opposite directions are called parallel vectors.

Question 7

According to the triangle law of vector addition, what is the sum of the directed line segments AB + BC?

  1. A.AC
  2. B.CA
  3. C.BA
  4. D.CB
Show the answer and why

Answer: A AC

Why

By the triangle law of vector addition, adding vector AB to vector BC gives the resultant vector AC, connecting initial point A directly to terminal point C.

Question 8

A vector has an initial point E = (2, 5) and a terminal point F = (2, 10). What is its magnitude |EF|?

  1. A.10
  2. B.7
  3. C.5
  4. D.25
Show the answer and why

Answer: C 5

Why

Using the distance formula: |EF| = √((2 - 2)2 + (10 - 5)2) = √(0 + 52) = √25 = 5.

Question 9

A vector in the first quadrant makes an angle of 37° with the positive horizontal axis (East). If North is along the positive vertical axis, how is its direction correctly written?

  1. A.N 37° E
  2. B.N 53° E
  3. C.E 53° N
  4. D.S 53° E
Show the answer and why

Answer: B N 53° E

Why

Vector direction starts from North or South, gives the angle measure, and ends with East or West. The complement of 37° with respect to North is 90° - 37° = 53°. Therefore, the direction is N 53° E.

Question 10

Vector CD has initial point C = (10, 1) and terminal point D = (5, 5). What is the magnitude |CD|?

  1. A.√41
  2. B.√26
  3. C.9
  4. D.√29
Show the answer and why

Answer: A √41

Why

Using the distance formula: |CD| = √((5 - 10)2 + (5 - 1)2) = √((-5)2 + 42) = √(25 + 16) = √41.

The app holds 12 questions on this unit, marked as you go. These are the first 10.

The textbook pages for unit 8

Unit 8 runs from page 247 to page 278 of the Ministry of Education edition.

Ministry of Education edition, English, 337 pages

Download the PDF

All Grade 9 Mathematics short notes

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