Vectors Summary

Vectors in a Plane

Vectors and Scalars

  • Vectors are used in physics to describe quantities with magnitude and direction (e.g., motion, force).

  • Scalars have magnitude only (e.g., time, mass, area).

  • Vectors have both magnitude and direction (e.g., displacement, velocity, force, acceleration).

  • Vectors are represented by a line segment with an arrow indicating direction; denoted by a symbol with a tilde (e.g., a⃗\vec{a}).

  • Two vectors are equal if their magnitude and direction are equal.

Addition of Vectors - Triangle Rule

  • If traveling from A to B (vector u⃗\vec{u}) and then from B to C (vector v⃗\vec{v}), the resultant vector w⃗\vec{w} from A to C is the sum: w⃗=u⃗+v⃗\vec{w} = \vec{u} + \vec{v}.

Negative of a Vector

  • If u⃗\vec{u} is the vector from A to B, then −u⃗-\vec{u} is the vector from B to A.

  • Subtracting vectors involves adding the negative of the second vector to the first.

Scalar Multiplication

  • Multiplying a vector by a scalar changes the magnitude.

  • If the scalar is negative, the direction is reversed.

  • Example: If vector u⃗\vec{u} has a magnitude of 10 and a direction of north, then 3u⃗3\vec{u} has a magnitude of 30 and a direction of north, while −2u⃗-2\vec{u} has a magnitude of 20 and a direction of south.

Magnitude and Direction

  • Pythagoras' theorem is used to find the magnitude of the resultant vector.

  • Trigonometry is used to find the direction (angle).

  • True bearings: Start at 0° and move clockwise.

  • Conventional bearings: N/S ° and then E/W.

Cartesian Form of a Vector

  • A vector can be represented in 2-dimensional planes

  • A position vector defines a point by magnitude and direction relative to the origin.

  • A vector u⃗\vec{u} can be expressed as u⃗=xi^+yj^\vec{u} = x\hat{i} + y\hat{j}, where i^\hat{i} and j^\hat{j} are unit vectors along the x and y axes, respectively.

Ordered Pair and Column Vector Notation

  • If a point C has coordinates (x, y), then u⃗=xi^+yj^\vec{u} = x\hat{i} + y\hat{j}.

  • Vector u⃗=xi^+yj^\vec{u} = x\hat{i} + y\hat{j} can be expressed as an ordered pair (x, y) or in column vector notation as (x y)\begin{pmatrix} x \ y \end{pmatrix}.

Converting from Cartesian to Polar Form

  1. Find the magnitude: ∣u⃗∣=x2+y2|\vec{u}| = \sqrt{x^2 + y^2}.

  2. Find the direction: tan(θ)=yxtan(\theta) = \frac{y}{x}; adjust the angle based on the quadrant.

  • Polar form: [r,θ][r, \theta], where r is the magnitude and θ\theta is the direction from the positive x-axis.

Converting from Polar to Cartesian Form

  • Resolve the vector into its x and y components: x=r⋅cos(θ)x = r \cdot cos(\theta), y=r⋅sin(θ)y = r \cdot sin(\theta).

Multiplying by a Scalar in Cartesian Form

  • If u⃗=xi^+yj^\vec{u} = x\hat{i} + y\hat{j}, then ku⃗=kxi^+kyj^k\vec{u} = kx\hat{i} + ky\hat{j}.

Scalar Multiplication in Polar Form

  • Multiplying by a positive scalar affects the magnitude only.

  • Multiplying by a negative scalar reverses the direction.

Position Vectors

  • The vector between two points A(x1, y1) and B(x2, y2) is given by: AB⃗=(x<em>2−x</em>1)i^+(y<em>2−y</em>1)j^\vec{AB} = (x<em>2 - x</em>1)\hat{i} + (y<em>2 - y</em>1)\hat{j}.

Unit Vectors

  • Unit vectors have a magnitude of 1.

  • A unit vector in the direction of u⃗\vec{u} is found by dividing the vector by its magnitude: n^=u⃗∣u⃗∣\hat{n} = \frac{\vec{u}}{|\vec{u}|}.