Vectors Def. A vector is a quantity that has both magnitude and direction. v is displacement vector from A to B A is the initial point, B is the terminal point v

Note that position was not used to determine a vector… v and u are equivalent vectors v = u

Def. If u and v are positioned so that the terminal point of u is at the initial point of v, then u + v is the vector with the initial point of u and the terminal point of v.

Def. If c is a scalar (non-vector) and v is a vector, then cv is the vector with the same direction as v that has length c times as long as v. If c < 0, then cv goes in the opposite direction as v.

u – v = u + (-v)

Ex. Given vectors a and b, draw 2a – b.

The vector from (0,0) to (3,4) can be written in component form The vector representation that has an initial point at (0,0) is called the position vector

Def. The magnitude of a vector is the distance between initial and terminal points. Thm. The length of vector is

Def. The direction angle of a vector is the angle between the vector and the positive x-axis. Ex. Find the direction angle of

Back to adding and scalar multiplication: When adding vectors in component form, add corresponding components When multiplying by a scalar, multiply each component by the scalar  Note that we haven’t talked about multiplying two vectors

Ex. Let, find a) |a| b) a + b c) 2a – 5b

We can also find the vector if we know magnitude and angle: Horizontal component Vertical component

Ex. The magnitude of a vector is 3, and it forms an angle of. Find the vector.

Def. A unit vector is a vector whose length is 1. The unit vector in the direction of a is Ex. Find the unit vector in the direction of

In physics, vectors can be used to describe force: If several forces act on an object, the resultant force is the sum of these forces. Ex. A boat is traveling due east in a river at a speed of 25 mph. If the river is moving 10 mph due north, what is the actual speed of the boat, and in what direction?

Ex. A boat traveling 30 mph has a compass heading of 100° east of north. The current velocity has a magnitude of 15 mph with a heading of 22° east of north. Find the resultant velocity of the boat.

Dot Product Def. Let and, then the dot product is This is also called the scalar product, since the result is a scalar.

Ex. Evaluate

Dot product is used to find the angle between two vectors: Thm. If θ is the angle between u and v, then Ex. Find the angle between and

Thm. Two vectors a and b are orthogonal if a ∙ b = 0. Orthogonal = Perpendicular = Normal Ex. Show that and are orthogonal.