10.4 vectors. a vector is a quantity that has both magnitude and direction. vectors in the plane can...
TRANSCRIPT
A vector is a quantity that has both magnitude 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.
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.
The direction of PQ is from P to Q.
The vector v whose magnitude is 0 is called the zero vector, 0.
Two vectors v and w are equal, written
v wif they have the same magnitude and direction.
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
An algebraic vector v is represented as
v = < a, b >
where a and b are real numbers (scalars) called the components of the vector v.
If v = < a, b > is an algebraic vector with initial point at the origin O and terminal point P = (a, b), then v is called a position vector.
Theorem
Suppose that v is a vector with initial point P1=(x1, y1), not necessarily the origin, and terminal point P2=(x2, y2). If v=P1P2, then v is equal to the position vector
Theorem Equality of Vectors
Two vectors v and w are equal if and only if their corresponding components are equal. That is,
Let i denote a unit vector whose direction is along the positive x-axis; let j denote a unit vector whose direction is along the positive y-axis. Any vector v = < a, b > can be written using the unit vectors i and j as follows:
Theorem Unit Vector in Direction of vFor any nonzero vector v, the vector
is a unit vector that has the same direction as v.