Chapter 1 · Vectors · Practice
Practice every vector concept
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Notation & components
A vector is an ordered list of numbers, written as a column: v = [3, 2]. Each entry is a component — one number per direction (x, y, z, ...).
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Write the vector that moves 4 units right and 3 units up as a column [x, y].
First number is the horizontal (x) move, second is vertical (y).
Right is +x, up is +y, so the vector is [4, 3].
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For v = [-2, 5], what is the x-component?
The x-component is the first entry.
Components are listed x then y, so the x-component is -2.
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A 3D vector has components x=1, y=0, z=-4. Write it as [x, y, z].
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Just place the three numbers in order.
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A car moves 6 m east then 6 m west. Write its displacement vector [x, y].
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East is +x, west is -x. They add along the same lane.
Magnitude
The length (magnitude) of v = [a, b] is ||v|| = sqrt(a^2 + b^2). In n dimensions, square every component, add, take one square root.
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Find the magnitude of v = [3, 4].
||v|| = sqrt(3^2 + 4^2).
sqrt(9 + 16) = sqrt(25) = 5.
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Find the magnitude of v = [0, -7].
A vector on an axis has length equal to the absolute value.
sqrt(0^2 + (-7)^2) = sqrt(49) = 7.
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Find ||v|| for v = [-4, 3]. (The minus sign should not change the length.)
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Squaring removes the sign: (-4)^2 = 16.
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Find the magnitude of v = [1, 1]. Give the exact value.
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sqrt(1 + 1).
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Find the magnitude of the 3D vector v = [2, 3, 6].
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sqrt(2^2 + 3^2 + 6^2).
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A vector [a, 12] has magnitude 13. Find the positive value of a.
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a^2 + 12^2 = 13^2. Solve for a.
Addition & subtraction
Add/subtract componentwise (lane by lane). v + w = [v1+w1, v2+w2]. v - w points from the tip of w to the tip of v.
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Compute [2, 1] + [1, 3] as [x, y].
Add lane by lane: x with x, y with y.
[2+1, 1+3] = [3, 4].
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Compute [5, -2] + [-3, 2] as [x, y].
Watch the signs in each lane.
[5+(-3), -2+2] = [2, 0].
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Compute [4, 3] - [1, 1] as [x, y].
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Subtract lane by lane.
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Compute [3, 2] - [1, 3] as [x, y].
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x: 3-1. y: 2-3 (goes negative).
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You walk [2,1] then [1,3]. What single vector gets you to the same spot?
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The single trip is the sum of the two.
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Find w so that [2, 5] + w = [6, 1]. Give w as [x, y].
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w = [6,1] - [2,5]. Subtract lane by lane.
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Compute [1,2,3] + [4,-2,-3] as [x, y, z].
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Three lanes now — add each separately.
Scalar multiplication
Multiply every component by the scalar c: c·v = [c·v1, c·v2]. c>1 stretches, 0<c<1 shrinks, c<0 flips direction.
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Compute 3 · [2, -1] as [x, y].
Multiply every component by 3.
[3·2, 3·(-1)] = [6, -3].
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Compute 0.5 · [4, 8] as [x, y].
Halve each component.
[0.5·4, 0.5·8] = [2, 4] — same direction, half the length.
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Compute -1 · [3, -2] as [x, y].
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Flip the sign of each component.
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Compute -2 · [1, -3] as [x, y].
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Multiply by -2: doubles the length and flips it.
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If c·[2, 3] = [8, 12], what is the scalar c?
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Compare one lane: c·2 = 8.
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Compute 2·[1,2] + 3·[0,-1] as [x, y].
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Scale each vector first, then add lane by lane.
Dot product
v · w = v1·w1 + v2·w2 + ... (one number). Also ||v|| ||w|| cos(theta). Zero means perpendicular; positive agree; negative oppose.
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Compute the dot product [3, 1] · [2, 4].
Multiply lane by lane, then add: 3·2 + 1·4.
3·2 + 1·4 = 6 + 4 = 10.
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Compute [1, 0] · [0, 1].
These are the standard basis vectors.
1·0 + 0·1 = 0. They are perpendicular.
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Compute [2, 1] · [-1, 2]. What does the result tell you about the angle?
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Compute the number; zero means perpendicular.
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Compute [2, 1] · [-2, -1].
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This is v dotted with a negative multiple of itself.
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Compute [1, 2, 3] · [4, 5, 6].
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Three lanes: 1·4 + 2·5 + 3·6.
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For which value of k are [3, k] and [2, -6] perpendicular?
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Perpendicular means dot product = 0: 3·2 + k·(-6) = 0.
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v = [3, 4] and w = [4, 0]. Using v·w = ||v|| ||w|| cos(theta), find cos(theta).
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v·w = 12, ||v|| = 5, ||w|| = 4. Divide.
Unit vectors
A unit vector has length 1. Normalize with v-hat = v / ||v||. The standard basis is i = [1,0], j = [0,1].
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What is the magnitude of any unit vector?
It is in the name.
A unit vector has length exactly 1.
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Normalize v = [3, 4]. Give the unit vector as [x, y].
Divide by ||v|| = 5.
||v|| = 5, so v-hat = [3/5, 4/5] = [0.6, 0.8].
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Normalize v = [0, -3]. Give the unit vector as [x, y].
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Length is 3; divide each component by 3.
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Write v = [3, 4] as ||v|| times a unit vector. Give the magnitude only.
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Every vector = magnitude × direction. Magnitude of [3,4]?
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Write [3, 2] as a combination of i=[1,0] and j=[0,1]: how many i's? (give the number)
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The x-component is the number of i-hats.
Applications
Velocity, force, displacement and data are vectors. Distance between two points a and b is ||b - a||.
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You are at (1,1); a cafe is at (4,5). Write the vector from you to the cafe [x, y].
Subtract: cafe - you.
[4-1, 5-1] = [3, 4].
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Using the previous problem's vector [3, 4], how far is the cafe (straight line)?
Distance = magnitude of the difference.
||[3,4]|| = sqrt(9+16) = 5.
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A boat rows [0, 3] while a current pushes [2, 0]. Write the actual velocity [x, y].
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Nature adds the two velocity vectors.
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Continuing: what is the boat's actual speed (magnitude of [2, 3])? Exact value.
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sqrt(2^2 + 3^2).
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Two forces [3, 1] and [-1, 4] act on a point. Write the net force [x, y].
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Net force is the sum of the force vectors.
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A word-meaning vector is [0.6, 0.8] and another is [0.8, 0.6]. Compute their dot product to gauge similarity.
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0.6·0.8 + 0.8·0.6.