2.1 Vectors in the Plane

← Chapter 2 §2.1 Vectors in the Plane 平面向量 · Chapter 2 · Vectors in Space

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Access for free at openstax.org · CC BY-NC-SA 4.0

	Property	Rule	As arrows
i	commutative	$\mathbf u+\mathbf v=\mathbf v+\mathbf u$	parallelogram
ii	associative	$(\mathbf u+\mathbf v)+\mathbf w=\mathbf u+(\mathbf v+\mathbf w)$	three legs, one trip
iii	additive identity	$\mathbf u+\mathbf 0=\mathbf u$	a walk of length 0
iv	additive inverse	$\mathbf u+(-\mathbf u)=\mathbf 0$	there and back
v	associativity of scalar multiplication	$r(s\mathbf u)=(rs)\mathbf u$	stretch twice
vi	distributive	$(r+s)\mathbf u=r\mathbf u+s\mathbf u$	stretches add along a line
vii	distributive	$r(\mathbf u+\mathbf v)=r\mathbf u+r\mathbf v$	scaled triangle
viii	identity and zero	$1\mathbf u=\mathbf u,\ \ 0\mathbf u=\mathbf 0$	—

Objective	Takeaways; say them first, then click
2.1.1 Describe a plane vector, using correct notation.	magnitude + direction; arrow $\overrightarrow{PQ}$ of length $\\|\mathbf v\\|$ ; equivalent = same length and direction
2.1.2 Perform basic vector operations.	$k\mathbf v$ stretches or flips; tip to tail; $\mathbf v-\mathbf w=\mathbf v+(-\mathbf w)$ ; slot by slot
2.1.3 Express a vector in component form.	head from the origin $\langle x,y\rangle$ ; terminal − initial; $\langle\\|\mathbf v\\|\cos\theta,\ \\|\mathbf v\\|\sin\theta\rangle$
2.1.4 Explain the formula for the magnitude of a vector.	$\sqrt{x^2+y^2}$ : Pythagoras = distance formula
2.1.5 Express a vector in terms of unit vectors.	$\mathbf v/\\|\mathbf v\\|$ ; $x\,\mathbf i+y\,\mathbf j$
2.1.6 Give two examples of vector quantities.	force (Example 2.9); velocity (Example 2.10)

Mistake	Result	Fix
Initial minus terminal	$\langle -4,2\rangle$ in Example 2.4: backwards	terminal minus initial
$(4,-2)$ for a vector	points and vectors mixed up	angle brackets (p96)
Adding magnitudes	450 lb instead of 446.6 lb	add components, then take the length
$\\|k\mathbf v\\|=k\\|\mathbf v\\|$ for $k<0$	a negative length	$\|k\|\,\\|\mathbf v\\|$
$\tan^{-1}(y/x)$ without the signs	30° for $-\sqrt3\,\mathbf i-\mathbf j$ ; true 210°	sketch; add 180° when $x<0$
Radian mode	about 210 lb in Example 2.9	degree mode
"Wind from the NE" drawn toward the NE	the plane drifts the wrong way	add 180°: it blows toward 225°
Dividing by $\|x\|+\|y\|$ to normalize	$\langle 1,2\rangle/3$ has length $\sqrt5/3$	divide by $\sqrt{x^2+y^2}$
$2\mathbf v-\mathbf w$ read as $2(\mathbf v-\mathbf w)$	$\langle -2,12\rangle$ instead of $\langle 1,10\rangle$	the scalar multiplies one vector

#	task	answer
2.1	$\overrightarrow{ST}$ , $S(3,-1)$ , $T(-2,3)$	$\langle -5,4\rangle$
2.2	sketch $2\mathbf w-\mathbf v$	on the grid $\langle 4,-8\rangle$
2.3	which are equivalent? (p95)	a, b, e
2.4	$(-4,-5)\to(-1,2)$	$\langle 3,7\rangle$
2.5	$\mathbf a=\langle 7,1\rangle$ , $\mathbf b$ : $(3,2)\to(-1,-1)$	$\\|\mathbf a\\|=5\sqrt2$ ; $\mathbf b=\langle -4,-3\rangle$ ; $3\mathbf a-4\mathbf b=\langle 37,15\rangle$
2.6	prove $\mathbf u+(-\mathbf u)=\mathbf 0$	$\langle x-x,\ y-y\rangle=\langle 0,0\rangle$
2.7	length 10 at 120°	$\langle -5,\ 5\sqrt3\rangle$
2.8	length 5, opposite to $\langle 9,2\rangle$	$-\tfrac5{\sqrt{85}}\langle 9,2\rangle$
2.9	$\langle 16,-11\rangle$ ; unit vector at 225°	$16\mathbf i-11\mathbf j$ ; $-\tfrac{\sqrt2}2\mathbf i-\tfrac{\sqrt2}2\mathbf j$
2.10	550 mph north, wind 50 mph from the NW	about 516 mph

#	task	answer	key step
1	$\overrightarrow{PQ}$	$\langle 2,2\rangle=2\mathbf i+2\mathbf j$	terminal minus initial
5	$\overrightarrow{PQ}+\overrightarrow{PR}$	$\langle 0,6\rangle$	add componentwise
7	$2\overrightarrow{PQ}-2\overrightarrow{PR}$	$\langle 8,-4\rangle$	scale, then subtract
11	unit vector, $(-1,-3)\to(2,1)$	$\langle\tfrac35,\tfrac45\rangle$	$\langle 3,4\rangle$ has length 5
13	$Q$ on the $y$ -axis, $\\|\mathbf v\\|=\sqrt5$	$Q(0,2)$	$\sqrt{1+y^2}=\sqrt5$
15	$\mathbf a=2\mathbf i+\mathbf j$ , $\mathbf b=\mathbf i+3\mathbf j$	$\langle 3,4\rangle$ ; $\langle 1,-2\rangle$ ; $\langle 4,2\rangle$ ; $\langle -1,-3\rangle$ ; $\langle 3,-1\rangle$	$5\le\sqrt5+\sqrt{10}$
17	$\\|-3\mathbf a+\mathbf b-4\mathbf i+\mathbf j\\|$	15	sum is $\langle 0,15\rangle$
27	length 7 along $\langle 3,-5\rangle$ (p105)	$\approx\langle 3.60,-6.00\rangle$	normalize, scale by 7
29	length 2 at 30°	$\langle\sqrt3,1\rangle$	$\langle 2\cos30^\circ,2\sin30^\circ\rangle$
33	length 10 at $5\pi/6$	$\langle -5\sqrt3,5\rangle$	quadrant II
35	angle of $5\sqrt2\,\mathbf i-5\sqrt2\,\mathbf j$	$7\pi/4$	$-45^\circ+360^\circ$
43	$D$ for parallelogram $ABCD$ (p106)	$D(6,1)$	$\overrightarrow{AD}=\overrightarrow{BC}$
49	45 lb and 52 lb at 25° (p107)	94.71 lb at 13.42°	force lab, preset Exercise 49
53	550 mph at N43°E, wind 25 mph toward N15°E (p108)	572.19 mph, N41.82°E	bearing → angle $90^\circ-b$

Vectors in the Plane

A vector carries a direction as well as a magnitude

Draw a vector as an arrow: length = magnitude, head = direction

To sketch $\overrightarrow{PQ}$ , start at $P$ and put the head at $Q$

Equivalent vectors have the same length and direction, wherever they start

A scalar stretches or flips an arrow, never turns it

Add tip to tail: $\mathbf v+\mathbf w$ runs from the tail of $\mathbf v$ to the head of $\mathbf w$

The parallelogram gives the same sum, so $\mathbf v+\mathbf w=\mathbf w+\mathbf v$

Magnitudes do not simply add: $\|\mathbf v+\mathbf w\|\le\|\mathbf v\|+\|\mathbf w\|$

Subtracting $\mathbf w$ means adding $-\mathbf w$

For a combination like $2\mathbf v-\mathbf w$ : scale first, then add tip to tail

Drag $\mathbf v$ and $\mathbf w$ : the arrows land where the components say

A vector from the origin is named by its head: $\langle x,y\rangle$

On a grid, equal vectors make the same moves across and up

To test equivalence, count each arrow's moves

Component form = terminal point minus initial point

In component form, vectors add and scale slot by slot

In components, vector arithmetic is number arithmetic in each slot

Vectors obey the same algebra rules as numbers

Each property holds because it holds for numbers in every slot

The triangle inequality comes down to a perfect square

Magnitude and angle give the components: the legs of a right triangle

Length 4 at $-45^\circ$ is $\langle 2\sqrt2,\ -2\sqrt2\rangle$

Length moves the head along a ray; the angle moves it round a circle

The magnitude is the hypotenuse: $\|\langle x,y\rangle\|=\sqrt{x^2+y^2}$

Divide a vector by its length to get its direction

For a chosen length in a given direction, scale the unit vector

Every vector is $x$ steps of $\mathbf i$ plus $y$ steps of $\mathbf j$

A unit vector at angle $\theta$ is $\cos\theta\,\mathbf i+\sin\theta\,\mathbf j$

One vector, four descriptions: all the same right triangle

Two non-parallel vectors build every plane vector

Forces add as arrows: 300 lb + 150 lb gives less than 450 lb

Put one force on the $x$ -axis, add components, then find length and angle

Velocities add too: a crosswind changes both speed and heading

Wind toward 225°: add components to get 454.17 mph, 3.57° south of west

Two forces add fully only when they point the same way

A vector is a right triangle you can slide

Common mistakes

Practice and answers