Introduction to Vectors and Scalars

Hello everyone! Today, we’re going to dive into an important concept in maths: multiplying a vector by a scalar.

What is a Vector?

A vector is a quantity that has both magnitude (how much) and direction (which way). You can think of it like an arrow: the length of the arrow shows the magnitude, and the point it’s pointing towards shows the direction.

For example, if I say “move 3 units to the right,” that’s a vector. It has a magnitude of 3 and is pointing right.

What is a Scalar?

A scalar is a simple quantity that only has magnitude and no direction. It’s just a number. For example, the number 5 is a scalar.

What Does it Mean to Multiply a Vector by a Scalar?

When you multiply a vector by a scalar, you are changing the magnitude (the length) of the vector while keeping its direction the same. If the scalar is greater than 1, the vector becomes longer. If the scalar is between 0 and 1, the vector gets shorter. If the scalar is negative, the vector flips direction.

Key Rules for Multiplying Vectors by Scalars

  1. Length Changes: The size of the vector changes according to the scalar.
  2. Direction Remains: The direction of the vector stays the same, unless the scalar is negative.
  3. Formula: If you have a vector v represented as ( \mathbf{v} = \begin{pmatrix} x \ y \end{pmatrix}
    ) and a scalar ( k ), then multiplying the vector by the scalar gives you: k \cdot \mathbf{v} = k \cdot \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} k \cdot x \ k \cdot y \end{pmatrix}

Examples

Example 1: Positive Scalar

Let’s say we have a vector ( \mathbf{v} = \begin{pmatrix} 2 \ 3 \end{pmatrix}

) and we multiply it by a scalar ( k = 4 ).

4 \cdot \mathbf{v} = 4 \cdot \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 4 \cdot 2 \ 4 \cdot 3 \end{pmatrix} = \begin{pmatrix} 8 \ 12 \end{pmatrix}

Here, the new vector ( \begin{pmatrix} 8 \ 12 \end{pmatrix}

) is longer but points in the same direction.

Example 2: Scalar Between 0 and 1

Now, let’s take the same vector ( \mathbf{v} = \begin{pmatrix} 2 \ 3 \end{pmatrix}

) and multiply it by a scalar ( k = 0.5 ).

0.5 \cdot \mathbf{v} = 0.5 \cdot \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} 0.5 \cdot 2 \ 0.5 \cdot 3 \end{pmatrix} = \begin{pmatrix} 1 \ 1.5 \end{pmatrix}

The new vector ( \begin{pmatrix} 1 \ 1.5 \end{pmatrix}

) is shorter, but still points in the same direction.

Example 3: Negative Scalar

Lastly, if we multiply the vector ( \mathbf{v} = \begin{pmatrix} 2 \ 3 \end{pmatrix}

) by a negative scalar ( k = -2 ):

-2 \cdot \mathbf{v} = -2 \cdot \begin{pmatrix} 2 \ 3 \end{pmatrix} = \begin{pmatrix} -2 \cdot 2 \ -2 \cdot 3 \end{pmatrix} = \begin{pmatrix} -4 \ -6 \end{pmatrix}

Now, the new vector ( \begin{pmatrix} -4 \ -6 \end{pmatrix}

) points in the opposite direction.

Tips and Tricks

  1. Visualisation: Draw arrows to represent vectors before and after multiplication to see how they change.
  2. Practice: Work on problems involving different scalars to see how they affect the vector.
  3. Direction Check: Always check the sign of the scalar to determine if the direction changes.

Practice Questions

Easy Level Questions

  1. Multiply the vector ( \begin{pmatrix} 1 \ 2 \end{pmatrix}
    ) by 3.
  2. What is ( 2 \cdot \begin{pmatrix} 4 \ 5 \end{pmatrix}
    )?
  3. Multiply ( \begin{pmatrix} 0 \ 1 \end{pmatrix}
    ) by 4.
  4. What happens when you multiply ( \begin{pmatrix} 3 \ 3 \end{pmatrix}
    ) by 0.5?
  5. Find ( -1 \cdot \begin{pmatrix} 5 \ 0 \end{pmatrix}
    ).
  6. Multiply ( \begin{pmatrix} 6 \ 2 \end{pmatrix}
    ) by 0.
  7. What is ( 10 \cdot \begin{pmatrix} 1 \ 1 \end{pmatrix}
    )?
  8. Find ( 3 \cdot \begin{pmatrix} 2 \ -1 \end{pmatrix}
    ).
  9. What is ( -3 \cdot \begin{pmatrix} 2 \ 4 \end{pmatrix}
    )?
  10. Multiply the vector ( \begin{pmatrix} 1 \ 3 \end{pmatrix}
    ) by 2.

Medium Level Questions

  1. If ( \mathbf{v} = \begin{pmatrix} 3 \ 4 \end{pmatrix}
    ), find ( 5 \cdot \mathbf{v} ).
  2. Calculate ( 0.25 \cdot \begin{pmatrix} 8 \ 4 \end{pmatrix}
    ).
  3. What is ( -2 \cdot \begin{pmatrix} 6 \ 7 \end{pmatrix}
    )?
  4. Multiply ( \begin{pmatrix} 5 \ -5 \end{pmatrix}
    ) by 3.
  5. Find ( 0.5 \cdot \begin{pmatrix} 10 \ 10 \end{pmatrix}
    ).
  6. What is ( 4 \cdot \begin{pmatrix} -1 \ 2 \end{pmatrix}
    )?
  7. Multiply ( \begin{pmatrix} 2 \ 2 \end{pmatrix}
    ) by -1.
  8. Find ( 3.5 \cdot \begin{pmatrix} 4 \ -4 \end{pmatrix}
    ).
  9. What is ( -0.5 \cdot \begin{pmatrix} 2 \ 6 \end{pmatrix}
    )?
  10. Calculate ( 6 \cdot \begin{pmatrix} 3 \ 1 \end{pmatrix}
    ).

Hard Level Questions

  1. If ( \mathbf{u} = \begin{pmatrix} 1 \ 2 \end{pmatrix}
    ) and ( \mathbf{v} = \begin{pmatrix} 2 \ 3 \end{pmatrix}
    ), find ( 2 \cdot \mathbf{u} + 3 \cdot \mathbf{v} ).
  2. Calculate ( -3 \cdot \begin{pmatrix} -2 \ -1 \end{pmatrix}
    + 2 \cdot \begin{pmatrix} 1 \ 0 \end{pmatrix}
    ).
  3. If ( k = -4 ) and ( \mathbf{v} = \begin{pmatrix} 5 \ 3 \end{pmatrix}
    ), what is ( k \cdot \mathbf{v} )?
  4. Find ( 2 \cdot \begin{pmatrix} -3 \ 1 \end{pmatrix}
    + 0.5 \cdot \begin{pmatrix} 4 \ -2 \end{pmatrix}
    ).
  5. What is ( 5 \cdot \begin{pmatrix} 0 \ -2 \end{pmatrix}
    ) and how does it compare to ( -1 \cdot \begin{pmatrix} 0 \ 2 \end{pmatrix}
    )?
  6. Calculate ( 3 \cdot \begin{pmatrix} -1 \ -3 \end{pmatrix}
    + 4 \cdot \begin{pmatrix} 2 \ 2 \end{pmatrix}
    ).
  7. If ( \mathbf{v} = \begin{pmatrix} 7 \ -5 \end{pmatrix}
    ) and ( k = -2 ), find ( k \cdot \mathbf{v} ) and describe the result.
  8. What is ( 0.5 \cdot \begin{pmatrix} 6 \ 8 \end{pmatrix}
    – 2 \cdot \begin{pmatrix} 1 \ 2 \end{pmatrix}
    )?
  9. If ( k = 3 ) and ( \mathbf{v} = \begin{pmatrix} 1 \ 1 \end{pmatrix}
    ), calculate ( 4 \cdot \mathbf{v} – k \cdot \mathbf{v} ).
  10. Find ( k \cdot \begin{pmatrix} 2 \ 4 \end{pmatrix}
    + \begin{pmatrix} 3 \ -1 \end{pmatrix}
    ) for ( k = -1 ).

Answers and Explanations

Easy Level Answers

  1. ( \begin{pmatrix} 3 \ 6 \end{pmatrix}
    )
  2. ( \begin{pmatrix} 8 \ 10 \end{pmatrix}
    )
  3. ( \begin{pmatrix} 0 \ 4 \end{pmatrix}
    )
  4. ( \begin{pmatrix} 1 \ 1.5 \end{pmatrix}
    )
  5. ( \begin{pmatrix} -5 \ 0 \end{pmatrix}
    )
  6. ( \begin{pmatrix} 0 \ 0 \end{pmatrix}
    )
  7. ( \begin{pmatrix} 10 \ 10 \end{pmatrix}
    )
  8. ( \begin{pmatrix} 6 \ -3 \end{pmatrix}
    )
  9. ( \begin{pmatrix} -6 \ -12 \end{pmatrix}
    )
  10. ( \begin{pmatrix} 2 \ 6 \end{pmatrix}
    )

Medium Level Answers

  1. ( \begin{pmatrix} 15 \ 20 \end{pmatrix}
    )
  2. ( \begin{pmatrix} 2 \ 1 \end{pmatrix}
    )
  3. ( \begin{pmatrix} -12 \ -14 \end{pmatrix}
    )
  4. ( \begin{pmatrix} 15 \ -15 \end{pmatrix}
    )
  5. ( \begin{pmatrix} 5 \ 5 \end{pmatrix}
    )
  6. ( \begin{pmatrix} -4 \ 8 \end{pmatrix}
    )
  7. ( \begin{pmatrix} -2 \ -2 \end{pmatrix}
    )
  8. ( \begin{pmatrix} 14 \ -14 \end{pmatrix}
    )
  9. ( \begin{pmatrix} -1 \ -3 \end{pmatrix}
    )
  10. ( \begin{pmatrix} 18 \ 6 \end{pmatrix}
    )

Hard Level Answers

  1. ( \begin{pmatrix} 11 \ 16 \end{pmatrix}
    )
  2. ( \begin{pmatrix} -6 \ -3 \end{pmatrix}
    )
  3. ( \begin{pmatrix} -20 \ -12 \end{pmatrix}
    )
  4. ( \begin{pmatrix} -5 + 2 \ 1 – 1 \end{pmatrix}
    = \begin{pmatrix} -3 \ 0 \end{pmatrix}
    )
  5. ( \begin{pmatrix} 0 \ -10 \end{pmatrix}
    ) vs. ( \begin{pmatrix} 0 \ -2 \end{pmatrix}
    )
  6. ( \begin{pmatrix} -3 \ -9 \end{pmatrix}
    + \begin{pmatrix} 8 \ 8 \end{pmatrix}
    = \begin{pmatrix} 5 \ -1 \end{pmatrix}
    )
  7. ( \begin{pmatrix} -14 \ 10 \end{pmatrix}
    ) (direction flips)
  8. ( \begin{pmatrix} 3 – 2 \ 4 – 4 \end{pmatrix}
    = \begin{pmatrix} 1 \ 0 \end{pmatrix}
    )
  9. ( \begin{pmatrix} 4 \ 4 \end{pmatrix}
    – \begin{pmatrix} 3 \ 3 \end{pmatrix}
    = \begin{pmatrix} 1 \ 1 \end{pmatrix}
    )
  10. ( \begin{pmatrix} -2 \ -4 \end{pmatrix}
    + \begin{pmatrix} 3 \ -1 \end{pmatrix}
    = \begin{pmatrix} 1 \ -5 \end{pmatrix}
    )

I hope this explanation helps you understand how to multiply a vector by a scalar! Remember, practice makes perfect, so keep working on those questions! If you have any questions, feel free to ask.