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Adding, subtracting and multiplying by a number

Space geometry and vectors, grade 12 (4 units) Lesson 8 of 20Learning

Adding, subtracting and multiplying by a number

Adding vectors

uvu+vABC

The triangle rule: when the second vector starts where the first ends, the sum goes from the start of the first to the end of the second:

AB→+BC→\displaystyle \overrightarrow{AB}+\overrightarrow{BC} =\displaystyle = AC→\displaystyle \overrightarrow{AC}

uvABCD

The parallelogram rule: when u⃗=AB→\displaystyle {\vec{u}}={\overrightarrow{AB}} and v⃗=AD→\displaystyle {\vec{v}}={\overrightarrow{AD}} leave the same point, their sum is the diagonal AC→\displaystyle \overrightarrow{AC} of the parallelogram ABCD\displaystyle ABCD: AD→=BC→\displaystyle {\overrightarrow{AD}}={\overrightarrow{BC}}, so this is the triangle rule again.

  • Several vectors in a chain: AB→+BC→+CD→=AD→\displaystyle {\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}}={\overrightarrow{AD}}.
  • A chain that comes back to its start adds up to the zero vector: AB→+BC→+CA→=0⃗\displaystyle {\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}}={\vec{0}}.
Subtracting vectors

Subtracting a vector means adding its opposite: v⃗−u⃗=v⃗+(−u⃗)\displaystyle {\vec{v}-\vec{u}}={\vec{v}+(-\vec{u})}.

When both vectors leave the same point, the difference goes from the end of the one taken away to the end of the other:

AD→−AB→\displaystyle \overrightarrow{AD}-\overrightarrow{AB} =\displaystyle = BD→\displaystyle \overrightarrow{BD}

In the parallelogram in the drawing: BD→=v⃗−u⃗\displaystyle {\overrightarrow{BD}}={\vec{v}-\vec{u}} – the other diagonal.

Multiplying a vector by a number

A scalar is a number. The vector tu⃗\displaystyle t\vec{u}:

  • its length is ∣t∣⋅∣u⃗∣\displaystyle |t|\cdot|\vec{u}|;
  • its direction is that of u⃗\displaystyle \vec{u} when t>0\displaystyle t>0, and the opposite when t<0\displaystyle t<0;
  • 0⋅u⃗=0⃗\displaystyle {0\cdot\vec{u}}={\vec{0}}, and (−1)⋅u⃗=−u⃗\displaystyle {(-1)\cdot\vec{u}}={-\vec{u}}.

The laws of the operations:

u⃗+v⃗\displaystyle \vec{u}+\vec{v} =\displaystyle = v⃗+u⃗\displaystyle \vec{v}+\vec{u}
(u⃗+v⃗)+w⃗\displaystyle (\vec{u}+\vec{v})+\vec{w} =\displaystyle = u⃗+(v⃗+w⃗)\displaystyle \vec{u}+(\vec{v}+\vec{w})
t⋅(s⋅u⃗)\displaystyle t\cdot(s\cdot\vec{u}) =\displaystyle = (t⋅s)⋅u⃗\displaystyle (t\cdot s)\cdot\vec{u}
(t+s)u⃗\displaystyle (t+s)\vec{u} =\displaystyle = tu⃗+su⃗\displaystyle t\vec{u}+s\vec{u}
t(u⃗+v⃗)\displaystyle t(\vec{u}+\vec{v}) =\displaystyle = tu⃗+tv⃗\displaystyle t\vec{u}+t\vec{v}

These are the laws of algebra, so an expression in vectors is simplified like an algebraic one: open the brackets and collect like terms. In the card, write u\displaystyle u for u⃗\displaystyle \vec{u}.

Example: vectors in a box

uvwABCDA′B′C′D′

In the box ABCDA′B′C′D′\displaystyle ABCDA'B'C'D' let AB→=u⃗\displaystyle {\overrightarrow{AB}}={\vec{u}}, AD→=v⃗\displaystyle {\overrightarrow{AD}}={\vec{v}}, AA′→=w⃗\displaystyle {\overrightarrow{AA'}}={\vec{w}}.
Every vector between two vertices is a chain of edges:

  • AC′→=AB→+BC→+CC′→=u⃗+v⃗+w⃗\displaystyle {\overrightarrow{AC'}}={\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CC'}}={\vec{u}+\vec{v}+\vec{w}}.
  • A′C→=A′A→+AC→=−w⃗+u⃗+v⃗\displaystyle {\overrightarrow{A'C}}={\overrightarrow{A'A}+\overrightarrow{AC}}={-\vec{w}+\vec{u}+\vec{v}}.
  • BD′→=BA→+AD→+DD′→=−u⃗+v⃗+w⃗\displaystyle {\overrightarrow{BD'}}={\overrightarrow{BA}+\overrightarrow{AD}+\overrightarrow{DD'}}={-\vec{u}+\vec{v}+\vec{w}}.
Exercise 1

Simplify.

a

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        Exercise 2

        uvABCDMN

        ABCD\displaystyle ABCD is a parallelogram, AB→=u⃗\displaystyle {\overrightarrow{AB}}={\vec{u}}, AD→=v⃗\displaystyle {\overrightarrow{AD}}={\vec{v}}.
        M\displaystyle M is the midpoint of the side BC\displaystyle BC, and N\displaystyle N the midpoint of the side CD\displaystyle CD.
        Express in terms of u⃗\displaystyle \vec{u} and v⃗\displaystyle \vec{v}:

        a
        AM→\displaystyle \overrightarrow{AM}

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          b
          AN→\displaystyle \overrightarrow{AN}

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            c
            MN→\displaystyle \overrightarrow{MN}

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              Exercise 3

              uvwABCDA′B′C′D′

              In the box ABCDA′B′C′D′\displaystyle ABCDA'B'C'D': AB→=u⃗\displaystyle {\overrightarrow{AB}}={\vec{u}}, AD→=v⃗\displaystyle {\overrightarrow{AD}}={\vec{v}}, AA′→=w⃗\displaystyle {\overrightarrow{AA'}}={\vec{w}}. Express in terms of u⃗\displaystyle \vec{u}, v⃗\displaystyle \vec{v}, w⃗\displaystyle \vec{w}:

              a
              B′D→\displaystyle \overrightarrow{B'D}

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                b
                C′A→\displaystyle \overrightarrow{C'A}

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  c
                  D′B→\displaystyle \overrightarrow{D'B}

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    Exercise 4

                    In the triangle ABC\displaystyle ABC, what is AB→+BC→+CA→\displaystyle \overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}? And what is AB→−CB→\displaystyle \overrightarrow{AB}-\overrightarrow{CB}? Explain.