Vectors

angle θ

angle θ

If A makes the ________ with the +x- axis, then its x- and y- components are Acosθ and Asinθ respectively (where A is the magnitude of A)

same direction

The product is a vector with twice the magnitude and, since the scalar is positive, the ________ is A.

55 miles

________ per hour is a scalar (like mass, work, energy power, temperature change)

Displacement

________ (which is the difference between the final and initial positions) is the prototypical example of a vector: A (________)= 4 miles (magnitude) to the north (direction)

Direction of kA

________= the same as A if k is positive or the opposite of A if k is negative.

dot product

The ________ or scalar product is a scalar that results from multiplying the magnitude of one vector with the magnitude of the component of another vector that is parallel to the first:

Vectors

________ obey the commutative law for addition.

Scalar multiplication

________: multiply each component by the scalar.

B

The vector sum A + ________ means the vector A followed by ________, while the vector sum ________ + A means the vector ________ followed by A.

angle θ

The direction of any vector, A, can be specified by the ________, it makes with the positive x- axis such than tan (θ)= A** ᵧ /** Aₓ.

If A makes the angle θ with the +x-axis, then its x

and y-components are Acosθ and Asinθ respectively (where A is the magnitude of A)

Vector addition is carried out with the "tip-to-tail" method

place the tail of one vector at the tip of the other vector, then connect the exposed tail to the exposed tip

Vector subtraction is carried out with the "tip-to-tail" method, but the negative second vector is added A

B = A + (-B)

Vectors can be expressed in terms of components using the unit basis vectors to indicate direction

i indicates the +x-direction, j indicated the +y-direction, and k indicates the +z-direction

Vector addition

add respective components

Vector subtraction

subtract the respective components

Scalar multiplication

multiply each component by the scalar

The magnitude of any vector, A, can be computed by applying the Pythagorean Theorem to the scalar components of the sector, Aᵧ and Aₓ

A = √(Aₓ)^2 + (Aᵧ)^2

The cross product is a vector that has magnitude that results from multiplying the magnitude of one vector by the magnitude of the component of another vector thats perpendicular to the first

| A x B| = ABsin(θ)