Vector

What are vectors?

Quantities that have both magnitude and direction.

Examples of common vectors.

Displacement, velocity, and force.

How are vectors graphically represented?

As arrows, where the length represents magnitude and the arrowhead indicates direction.

2D representation of a vector A.

$$\textbf{A} = A_x \textbf{i} + A_y \textbf{j}$$

3D representation of a vector A.

$$\textbf{A} = A_x \textbf{i} + A_y \textbf{j} + A_z \textbf{k}$$

How is vector addition performed?

By adding vectors tip-to-tail or using components: $\textbf{A} + \textbf{B} = (A_x + B_x) \textbf{i} + (A_y + B_y) \textbf{j}$.

How is vector subtraction performed?

By using components: $\textbf{A} - \textbf{B} = (A_x - B_x) \textbf{i} + (A_y - B_y) \textbf{j}$.

What is scalar multiplication of a vector?

For a scalar $k$ and vector A: $k \textbf{A} = kA_x \textbf{i} + kA_y \textbf{j}$.

Dot Product of vectors A and B.

$\textbf{A} \bullet \textbf{B} = \| \textbf{A}\| \| \textbf{B}\| \cos \theta$, a scalar quantity.

Cross Product of vectors A and B.

$\textbf{A} \times \textbf{B} = \| \textbf{A} \| \| \textbf{B}\| \sin \theta \text{n}$, a vector perpendicular to both A and B.

Applications of vectors.

Used in mechanics to represent forces, velocities, and in solving problems of projectile and circular motion.

Characteristics of equal vectors.

Same magnitude and direction.

What is a unit vector?

A vector of magnitude 1.

What is a null vector?

A vector with zero magnitude; its direction is undefined.

What is a position vector?

Indicates the position of a point in space relative to a reference.

What does a velocity vector represent?

The rate of change of position with respect to time.

What does an acceleration vector represent?

The rate of change of velocity with respect to time.

What is the Parallelogram Law of Vector Addition?

If two vectors are represented as adjacent sides of a parallelogram, their resultant is given by the diagonal.

What is the Triangle Law of Vector Addition?

If two vectors are represented as two sides of a triangle taken in sequence, the resultant is the closing side of the triangle.