Honors Pre-Calculus 12.1 Vectors Page 419 Objective: Perform basic operations on vectors.

A vector is a quantity that has both magnitude (size) and direction. Vectors in the plane can be represented by arrows. The length of the arrow represents the magnitude of the vector. The arrowhead indicates the direction of the vector.

P Q Initial Point Terminal Point Directed line segment

If a vector v has the same magnitude and the same direction as the directed line segment PQ, then we write v = PQ The magnitude of the directed line segment PQ is the distance from point P to the point Q. If v is a vector, |v| represents the magnitude or numeric value of v.

The vector v whose magnitude is 0 is called the zero vector. Two vectors v and w are equal, written if they have the same magnitude and direction. The vector u whose magnitude is 1 is called the unit vector.

v = w w v

The negative of the vector v is a vector whose magnitude is the same but the direction is opposite. Two vectors v and z are opposite, written -v = z if they have the same magnitude (|v| = |z|) and opposite direction. The sum of v and z = 0.

-v = z z v

Vector addition is commutative. Vector addition is associative. v + w = w + v v + (u + w) = (v + u) + w v + 0 = 0 + v =v v + (-v) = 0

Multiplying Vectors by Numbers

Properties of Scalar Products

Initial point of v Terminal point of w v + w v w A B C

2v-w v w Use the vectors illustrated on the left to graph the expression. 2v -w

Vectors are often used in Physics to represent the combination of forces. The vector sum of forces is called the resultant of forces. This is typically measured in newton (N), which is the force needed to bring a 1 kg mass to a speed of 1 meter per second in 1 second.

Make a scale drawing showing a force of 20N pulling an object east and another force of 10N pulling the object in the compass direction of 150°. Draw the resultant force vector and find its magnitude and direction. 20N 10N 0˚North NO LONGER N

TB p. 423 #4, 11, 16, 25 (Do not have to do protractor)