A vector is a quantity that has both magnitude (size) and direction.
Think of it like an arrow: the length shows how much, and the direction shows which way!
Vector addition follows the “tip-to-tail” rule.
Place the tail of the second vector at the tip of the first vector. The result connects the tail of the first to the tip of the second!
Vector subtraction: A – B = A + (-B)
To subtract B from A, we add the negative of B (flip B’s direction) to A!
The dot product measures how much two vectors point in the same direction.
Result is a scalar (number), not a vector. Used for finding angles between vectors!
In 2D, cross product gives the area of the parallelogram formed by two vectors.
Result is a scalar in 2D. In 3D, it gives a vector perpendicular to both input vectors!
A force of 5N acts eastward, and another force of 3N acts northward. Find the resultant force.
Solution:
East = (5, 0), North = (0, 3)
Resultant = (5, 0) + (0, 3) = (5, 3)
Magnitude = √(5² + 3²) = √34 = 5.83N
Direction = arctan(3/5) = 31° north of east
Two vectors A = (4, 3) and B = (2, -1). Find the angle between them.
Solution:
A • B = (4)(2) + (3)(-1) = 8 – 3 = 5
|A| = √(4² + 3²) = 5
|B| = √(2² + (-1)²) = √5
cos(θ) = (A • B)/(|A||B|) = 5/(5√5) = 1/√5
θ = arccos(1/√5) = 63.4°