Vectors and Scalars in Physics

Vectors and Scalars

• A scalar is a quantity with only magnitude, like a normal number, and speed is an example of a scalar.
• A vector is a quantity with both magnitude and direction, and velocity is an example of a vector.
• Distance is a scalar that measures the length of the path traveled, while displacement is a vector that measures how far an object is from its starting point.

Vectors as Arrows

• Vectors can be represented as arrows and are often denoted with an arrow over the letter.
• Two vectors are equal if and only if they have the same direction and magnitude.
• A vector can be multiplied by a scalar, which scales its length.
• Vectors can be added "tip to tail".

The i-j Plane

• The i-j plane is a way to express a vector using only numbers by splitting the plane into two perpendicular axes, i and j.
• Every vector in a 2D plane can be broken down into its i and j components, which describe how much it goes across (i direction) and how much it goes up (j direction).
• Resolving a vector into its components is a way to describe it, and the opposite of this is finding the resultant vector.

The Resultant Vector and Vector Forms

• The resultant vector is the result of adding a vector's i and j components.
• There are two ways to describe a vector: rectangular form (using i and j) and polar form (using magnitude and direction).

The Dot Product

• The dot product is a way to multiply vectors, and it gives a scalar answer.
• The dot product is commutative, meaning the order of the vectors does not matter.
• The dot product can be used to find the angle between two vectors using the formula: a · b = |a| |b| cos(θ).
• If two vectors are perpendicular, their dot product is equal to 0, because cos(90) = 0.