Vectors in the Plane

Vectors #

A size that carries a direction too.

A vector carries a direction as well as a size

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a = (3, 2) means 3 across and 2 up. A vector carries a direction, not just a size.

Start the next arrow where a finished: b = (2, 3), 2 across and 3 up.

The parts add: 3 + 2 = 5 across, 2 + 3 = 5 up, so a + b = (5, 5).

Column Vectors #

Steps across and steps up, written down.

A column vector records a movement as steps across and steps up

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The vector goes 4 across and 3 up. A column vector writes the horizontal step on top.

Negating a vector keeps its length and reverses its direction.

Adding Vectors #

Follow one journey, then the next.

Adding vectors means following one after the other, so the components add

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Two vectors: a steps 3 across and 1 up, b steps 2 across and 4 up.

Adding means doing one then the other, so start b where a finished.

You land at (5, 5): 3 + 2 = 5 across and 1 + 4 = 5 up. Each part adds.

Subtracting Vectors #

The gap between two arrows: add the reverse.

Subtracting a vector means adding its reverse, so each component subtracts

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Subtracting is adding the reverse: negate each component of b, then add.

Drawn from one start, a − b is the arrow from b’s tip across to a’s tip.

The gap between two points is exactly this move: destination minus start.

Scalar Multiplication #

Copies of one journey: every step scales.

A scalar times a vector scales every component and keeps it on the same line

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The scalar 3 makes three copies of one journey: 3a takes (2, 1) out to (6, 3).

Same direction, three times the length — each component scales by three.

A negative number scales and flips — the reversing move from column vectors.

Every multiple of a lies on a’s own line — this is what collinear means.

Magnitude of a Vector #

Pythagoras on the two components.

The magnitude of a vector is Pythagoras applied to its two components

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Three across and four up draws a right triangle with the vector as its hypotenuse.

So its magnitude — its length — is the square root of 9 plus 16, which is 5.

Unit Vectors #

Divide a vector by its length: direction, size one.

Dividing a vector by its own length leaves the direction with a length of one

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Pythagoras gives this vector its length: 3 across and 4 up makes 5.

Divide both components by that length: three fifths across, four fifths up.

The short arrow points exactly the same way, and its length is now one.

The Unit Vectors i and j #

The same vector, spelled with letters.

The unit vectors i and j spell any vector as steps across plus steps up

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Two unit vectors on the grid: i is one step across, j is one step up.

Three steps of i and four of j — the same arrow, spelled with letters.

Arithmetic carries straight over: add the i parts, then add the j parts.

Elsewhere i names the imaginary unit — a different i; the context tells you which.

Position Vectors #

The journey to a point from the origin.

A position vector fixes a point by giving the journey to it from the origin

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The journey from the origin to A is the position vector of A.

B has one too. To get from A to B, undo the journey OA, then follow OB.

That is AB = OB − OA — the arrow straight from A to B.

In numbers: OB − OA = (2 − 5, 6 − 2) = (−3, 4), so 3 back and 4 up.

Vector Proofs with Midpoints #

Midpoints and ratios as short algebra.

Position vectors turn midpoint, ratio and parallel claims into short algebra

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M sits halfway from A to B. What is the journey from O straight out to M?

Walk to A, then half the gap — the algebra simplifies to half of a plus b.

Two parts of three sit behind P: start at a, then walk two thirds of the gap.

To prove PQ parallel to AB, show one is a multiple of the other. That is all.

The Ratio Theorem #

Any point on AB from its ratio, in one formula.

If P divides AB in the ratio $\lambda$λ : $\mu$μ, then OP $=\frac{\mu a+\lambda b}{\lambda +\mu }$= (μa + λb)/(λ + μ), each end weighted by the far part

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P divides AB in the ratio 2 : 1: two parts lie from A to P, and one from P to B.

Collect the terms and OP $=\frac{a+2b}{3}$= (a + 2b)/3: each end is weighted by the part on the far side of P.

This is the ratio theorem: if P divides AB in the ratio $\lambda$λ : $\mu$μ, then OP $=\frac{\mu a+\lambda b}{\lambda +\mu }$= (μa + λb)/(λ + μ).

Collinear Vectors #

Multiples of each other lie on one line.

Two vectors that are multiples of each other point along the same straight line

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a = (2, 1) and 3a = (6, 3): 6 = 3 × 2 and 3 = 3 × 1, one multiplier for both.

Now compare b = (3, 4) with a = (2, 1): 3 is 1.5 × 2, but 4 is 4 × 1.

No single multiplier fits both components, so a and b are not collinear.

The Dot Product #

Match the components, multiply, and add.

Multiplying matching components and adding measures how far two vectors point the same way

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Multiply matching components — across with across, up with up — then add.

It is a’s length times the part of b lying along a — biggest when the two line up.

At a right angle, b has no part lying along a at all, so the dot product is zero.

The Angle Between Vectors #

Divide the dot product by both lengths.

Dividing the dot product by both lengths leaves $\cos \theta$cos θ, the angle between the arrows

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$a\cdot b=|a||b|\cos \theta$a · b = |a| |b| cos θ, so dividing by both lengths leaves $\cos \theta$cos θ on its own.

A dot product of zero forces $\cos \theta =0$cos θ = 0 — and that angle is a right angle.

Pointing the same way, $\cos \theta =1$cos θ = 1 and the angle is zero: no turn at all.

$\cos \theta$cos θ is nearly 1: 24 over 25 means the two arrows point almost the same way.

Vectors in Three Dimensions #

A third component, one more square.

A third component points into depth, and the length just gains one more square

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A third number for depth: (2, 3, 6) cuts corner to corner through a box.

The length just gains one more square: the root of 4 + 9 + 36 is 7.

Seen in the box: that corner-to-corner diagonal measures exactly 7.

A third unit vector k steps into depth: 2i plus 3j plus 6k.

Adding is unchanged — three places now, and each one adds on its own.

Distance and Midpoint in Three Dimensions #

Pythagoras twice, and one average per axis.

A third coordinate adds a third square to the distance and a third average to the midpoint

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Two points in space are opposite corners of a box, and d is its long diagonal.

Pythagoras runs twice: once across the floor, then once up from that diagonal.

A(1, 2, 3) to B(7, 10, 27): the gaps are 6, 8 and 24, so the distance is 26.

The midpoint averages each coordinate on its own, z included: (4, 6, 15).

Check it: the midpoint stands 13 from each end, exactly half of 26.

The Vector Equation of a Line #

Start at a, walk t strides of d.

The equation r = a + t d names every point of a line by strides of d from a

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Start at a, then walk copies of d — every stride lands on the same line.

r = a + t d: pick a number t, and out comes a point. Every t names one.

t = 0 stays at a. t = 2 stands exactly two strides out along the line.

Doubling d draws the same line again — any multiple of d gives the same direction.