Vectors

Unit 7 Vectors

Key unit competence: Examine vectors in the plane in solving related problems.

Key ideas

• A vector has both magnitude (size) and direction; a scalar has only magnitude (mass, time). Displacement, velocity and force are vectors.
• A vector from $A$ to $B$ is written $\overrightarrow{AB}$. In the plane it is written as a column vector $\left(\begin{array}{c}x\\ y\end{array}\right)$: $x$ steps right (negative: left), $y$ steps up (negative: down).
• If $A(x_1,y_1)$ and $B(x_2,y_2)$, then $\overrightarrow{AB}=\left(\begin{array}{c}{x}_2-x_1\\ y_2-y_1\end{array}\right)$. The position vector of $A$ is $\overrightarrow{OA}=\left(\begin{array}{c}{x}_1\\ y_1\end{array}\right)$.
• Special vectors: the null vector $\left(\begin{array}{c}0\\ 0\end{array}\right)$; unit vectors $\vec{i}=\left(\begin{array}{c}1\\ 0\end{array}\right)$, $\vec{j}=\left(\begin{array}{c}0\\ 1\end{array}\right)$; the opposite of $\vec{u}$ is $-\vec{u}$ ($\overrightarrow{BA}=-\overrightarrow{AB}$).
• Equal vectors have the same magnitude and direction (same components). Addition: triangle law $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$; add components.
• Scalar multiplication: $k\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}kx\\ ky\end{array}\right)$. If $k<0$ the direction is reversed.
• Magnitude: $\left|\left(\begin{array}{c}x\\ y\end{array}\right)\right|=\sqrt{x^2+y^2}$. The distance $AB=|\overrightarrow{AB}|$. Midpoint of $AB$: $M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

Formulas to know

Common misconceptions

• "$\overrightarrow{AB}=\overrightarrow{BA}$." No: they have the same length but opposite directions: $\overrightarrow{BA}=-\overrightarrow{AB}$.
• "$\overrightarrow{AB}$ = coordinates of $A$ minus coordinates of $B$." No: it is end minus start: $B-A$.
• "$|\vec{u}+\vec{v}|=|\vec{u}|+|\vec{v}|$." Not in general; it is only true when the vectors have the same direction.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. Given $P(3,-1)$ and $Q(-1,2)$, find $\overrightarrow{PQ}$ and $|\overrightarrow{PQ}|$.

   Show answer

   $\overrightarrow{PQ}=\left(\begin{array}{c}-4\\ 3\end{array}\right)$, $|\overrightarrow{PQ}|=5$.

   Common mistake: Subtracting start minus end.

2. The midpoint of $PQ$ is $(2,-1)$ and $P(5,3)$. Find $Q$.

   Show answer

   $Q=(2\times 2-5,2\times (-1)-3)=(-1,-5)$.

   Common mistake: Averaging $P$ and $M$ instead.

3. Given $\vec{u}=\left(\begin{array}{c}4\\ -2\end{array}\right)$ and $\vec{v}=\left(\begin{array}{c}1\\ 5\end{array}\right)$, find $\vec{u}+\vec{v}$, $\vec{u}-\vec{v}$ and $-2\vec{v}$.

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   $\left(\begin{array}{c}5\\ 3\end{array}\right)$, $\left(\begin{array}{c}3\\ -7\end{array}\right)$, $\left(\begin{array}{c}-2\\ -10\end{array}\right)$.

   Common mistake: Multiplying only one component by $-2$.

4. Write $\left(\begin{array}{c}-3\\ 7\end{array}\right)$ in terms of $\vec{i}$ and $\vec{j}$.

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   $-3\vec{i}+7\vec{j}$.

   Common mistake: Reversing the components.

5. Find the midpoint of $A(-4,6)$ and $B(2,-2)$.

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   $(-1,2)$.

   Common mistake: Subtracting the coordinates instead of adding.

6. A man walks with displacement $\left(\begin{array}{c}5\\ 0\end{array}\right)$ km then $\left(\begin{array}{c}0\\ 12\end{array}\right)$ km. Find his total displacement and its magnitude.

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   $\left(\begin{array}{c}5\\ 12\end{array}\right)$, magnitude 13 km.

   Common mistake: Adding the distances (17 km).

7. Simplify $\overrightarrow{AB}-\overrightarrow{CB}$.

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   $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$.

   Common mistake: Writing $\overrightarrow{AC}$ with the wrong order of letters.

8. Show that $A(1,1)$, $B(4,2)$, $C(5,5)$, $D(2,4)$ form a parallelogram.

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   $\overrightarrow{AB}=\left(\begin{array}{c}3\\ 1\end{array}\right)$ and $\overrightarrow{DC}=\left(\begin{array}{c}3\\ 1\end{array}\right)$. Equal vectors, so $AB$ and $DC$ are parallel and equal: $ABCD$ is a parallelogram.

   Common mistake: Comparing $\overrightarrow{AB}$ with $\overrightarrow{CD}$ (opposite vectors).

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Find $k$ if $\left|\left(\begin{array}{c}k\\ 12\end{array}\right)\right|=13$.	a. 10
   2. $\vec{u}=\left(\begin{array}{c}2\\ -1\end{array}\right)$ and $\vec{v}=\left(\begin{array}{c}-3\\ 4\end{array}\right)$. Find $2\vec{u}-3\vec{v}$.	b. $\left(\begin{array}{c}13\\ -14\end{array}\right)$
   3. Find the distance between $A(-1,3)$ and $B(5,-5)$.	c. $k=\pm 5$
   4. $M$ is the midpoint of $AB$ with $A(1,1)$ and $M(3,4)$. Find $B$.	d. $(5,7)$

   Show answer

   1 → c ($k=\pm 5$) · 2 → b ($\left(\begin{array}{c}13\\ -14\end{array}\right)$) · 3 → a (10) · 4 → d ($(5,7)$)

   • Step 1. $k^2+144=169$, so $k^2=25$.
     Step 2. $k=5$ or $k=-5$.
   • Step 1. $2\vec{u}=\left(\begin{array}{c}4\\ -2\end{array}\right)$ and $3\vec{v}=\left(\begin{array}{c}-9\\ 12\end{array}\right)$.
     Step 2. Subtract: $\left(\begin{array}{c}4+9\\ -2-12\end{array}\right)=\left(\begin{array}{c}13\\ -14\end{array}\right)$.
   • $\overrightarrow{AB}=\left(\begin{array}{c}6\\ -8\end{array}\right)$, length $\sqrt{36+64}=10$.
   • $B=2M-A=(6-1,8-1)$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $x=5,y=3$  ·  $\left(\begin{array}{c}6\\ -3\end{array}\right)$  ·  $\left(\begin{array}{c}5\\ 2\end{array}\right)$  ·  twice as long
   1. $|2\vec{u}|$ compared with $|\vec{u}|$ is .
   2. If $\vec{u}=\left(\begin{array}{c}x\\ 3\end{array}\right)$ equals $\left(\begin{array}{c}5\\ y\end{array}\right)$, then .
   3. Find $3\left(\begin{array}{c}2\\ -1\end{array}\right)$. →
   Show answer

   1: twice as long · 2: $x=5,y=3$ · 3: $\left(\begin{array}{c}6\\ -3\end{array}\right)$

   • $|k\vec{u}|=|k||\vec{u}|$.
   • Equal vectors have equal components.
   • Multiply each component by 3.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. If $\overrightarrow{AB}=\left(\begin{array}{c}2\\ -5\end{array}\right)$, then $\overrightarrow{BA}=$	a. $(4,7)$
   2. Find the midpoint of $A(2,4)$ and $B(6,10)$.	b. $\left(\begin{array}{c}-2\\ 5\end{array}\right)$
   3. Find $a$ and $b$ if $a\left(\begin{array}{c}1\\ 3\end{array}\right)+b\left(\begin{array}{c}2\\ -1\end{array}\right)=\left(\begin{array}{c}8\\ 3\end{array}\right)$.	c. $\left(\begin{array}{c}3\\ -2\end{array}\right)$
   4. Write $3\vec{i}-2\vec{j}$ as a column vector.	d. $a=2,b=3$

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   1 → b ($\left(\begin{array}{c}-2\\ 5\end{array}\right)$) · 2 → a ($(4,7)$) · 3 → d ($a=2,b=3$) · 4 → c ($\left(\begin{array}{c}3\\ -2\end{array}\right)$)

   • Reverse the direction: change both signs.
   • $\left(\frac{2+6}{2},\frac{4+10}{2}\right)$.
   • Step 1. $a+2b=8$ and $3a-b=3$, so $b=3a-3$.
     Step 2. $a+6a-6=8\Rightarrow a=2$, $b=3$.
   • Coefficient of $\vec{i}$ on top, of $\vec{j}$ below.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $(-3,4)$  ·  5  ·  $(2,3)$  ·  10 km/h
   1. Find the magnitude of $\left(\begin{array}{c}3\\ 4\end{array}\right)$. →
   2. Which point is 5 units from $(0,0)$? →
   3. A boat travels with velocity $\left(\begin{array}{c}6\\ 8\end{array}\right)$ km/h. What is its speed? →
   Show answer

   1: 5 · 2: $(-3,4)$ · 3: 10 km/h

   • $\sqrt{9+16}=5$.
   • $\sqrt{9+16}=5$.
   • Speed is the magnitude: $\sqrt{36+64}=10$.

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