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13.1 Vectors

scalars: numbers such as \( 5, \pi, e, 133, \text{etc} \)

magnitude only, no direction

vectors: magnitude and direction

e.g. wind 25 mph from NE

vectors also tell us the relative position from one point to another

for example, \( P(1, 2) \) to \( Q(3, -4) \)

Figure: A coordinate plane showing a vector arrow from point P(1,2) to point Q(3,-4) in the fourth quadrant.

vector from P to Q

\[ \vec{PQ} = \langle 3-1, -4-2 \rangle = \langle 2, -6 \rangle \]

2 unit RIGHT (positive)

6 units down (negative)

\( Q \) to \( P \)

\[ \vec{QP} = \langle -2, 6 \rangle \]\[ \vec{QP} = -\vec{PQ} \]

minus sign reverses direction

Figure: Two parallel vectors pointing in opposite directions, labeled vector a and negative vector a.

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magnitude / length

\( \vec{u} = \langle a, b \rangle \)

\[ \text{length} : |\vec{u}| = \sqrt{a^2 + b^2} \]

Figure: A right triangle diagram showing vector u as the hypotenuse with horizontal component a and vertical component b.

\( \vec{PQ} = \langle -2, 6 \rangle \)

\[ |\vec{PQ}| = \sqrt{(-2)^2 + (6)^2} = \sqrt{40} \]

addition / subtraction

\( \vec{u} = \langle 1, 2 \rangle \quad \vec{v} = \langle 0, 3 \rangle \)

\[ \vec{u} + \vec{v} = \langle 1+0, 2+3 \rangle = \langle 1, 5 \rangle \]\[ \vec{u} - \vec{v} = \langle 1-0, 2-3 \rangle = \langle 1, -1 \rangle \]\[ 2\vec{u} - 3\vec{v} = 2\langle 1, 2 \rangle - 3\langle 0, 3 \rangle \]\[ = \langle 2, 4 \rangle - \langle 0, 9 \rangle \]\[ = \langle 2, -5 \rangle \]

Figure: Three parallel vectors of different lengths and directions, labeled vector a, 2 times vector a, and negative 2 times vector a.

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Graphical Vector Addition

Individual Vectors

Graphically, we represent vectors as directed line segments. Here are two vectors, \( \vec{u} \) and \( \vec{v} \), shown independently.

Figure: Two separate vectors: vector u points up and to the right, and vector v points vertically upward.

Vector Addition: \( \vec{u} + \vec{v} \)

"Triangle Rule"

To add vectors using the triangle rule, place the tail of the second vector at the head of the first. The resultant vector \( \vec{u} + \vec{v} \) is the vector from the tail of the first to the head of the second.

"Parallelogram Rule"

Alternatively, place both vectors at the same starting point. The resultant vector \( \vec{u} + \vec{v} \) is the diagonal of the parallelogram formed by the two vectors.

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Unit Vectors

Unit vector: vector of length or magnitude of 1

\[ \vec{u} = \langle 3, 2 \rangle \quad |\vec{u}| = \sqrt{3^2 + 2^2} = \sqrt{13} \neq 1 \]

So, \( \vec{u} \) is NOT a unit vector.

But

\[ \vec{v} = \frac{\vec{u}}{|\vec{u}|} = \frac{\langle 3, 2 \rangle}{\sqrt{13}} = \left\langle \frac{3}{\sqrt{13}}, \frac{2}{\sqrt{13}} \right\rangle \]

\[ |\vec{v}| = \sqrt{\left(\frac{3}{\sqrt{13}}\right)^2 + \left(\frac{2}{\sqrt{13}}\right)^2} = \sqrt{\frac{9}{13} + \frac{4}{13}} = \sqrt{\frac{13}{13}} = 1 \]

So, in general, \( \frac{\vec{a}}{|\vec{a}|} \) is a unit vector.

Directional Unit Vectors

Unit vector in opposite direction of \( \vec{b} \)?

\[ -\frac{\vec{b}}{|\vec{b}|} \]

Vector length of 3, in opposite direction of \( \vec{b} \)?

\[ -3\frac{\vec{b}}{|\vec{b}|} \]

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Special Unit Vectors

\(\vec{i} = \langle 1, 0 \rangle\): unit vector in \(x\) direction
\(\vec{j} = \langle 0, 1 \rangle\): unit vector in \(y\) direction

Figure: 2D Cartesian axes with unit vector i on the x-axis and unit vector j on the y-axis.

13.2 Vectors in 3D

Given points \(P(1, 2, 3)\) and \(Q(4, 6, 8)\):

\[\vec{PQ} = \langle 4-1, 6-2, 8-3 \rangle = \langle 3, 4, 5 \rangle = 3\vec{i} + 4\vec{j} + 5\vec{k}\]

Note: \(\vec{k} = \langle 0, 0, 1 \rangle\)

Shapes

Line in 2D: \(y = 5\)

All points with \(y = 5\)

Figure: 2D graph with a horizontal dashed line passing through y=5, labeled 'all points w/ y=5'.

In 3D, it's a plane: \(z = 5\)

All points with \(z = 5\), represented as \((x, y, 5)\)

Figure: 3D coordinate system showing a horizontal plane at height z=5, labeled 'all points w/ z=5'.