Physics and Math 1.2 Vectors and Scalars

cos 0°

1

sin 90°

1

tan 45°

1

Right-hand rule

1. Start by pointing your thumb in the direction of vector A. 2. Extend your fingers in the direction of vector B. 3. Your palm establishes the plane between the two vectors. The direction your palm points is the direction of the resultant C.

cos 60°

1/2

sin 30°

1/2

Dot product equation

A ° B = |A||B|cosθ

Find the x- and y-components of the following vector: V = 10 m/s and θ = 30°.

CAH: Y = V cosθ = (10 m/s) sin(30°) = 10(√3)/2 = 5√3 SOH: X = V sinθ = (10 m/s) sin(30°) = 10(1/2) = 5

What is the magnitude and direction of the vector with the following components: X = 3 m/s and Y = 4 m/s ?

V = √X² +Y² =√3² + 4² = √9 + 16 = √25 = 5

Vectors

are physical quantities that have both magnitude and direction. Vector quantities include displacement, velocity, acceleration, and force, among others.

Component vectors equation

SOH CAH TOA. In other words: X = V cosθ Y = V sinθ Where V is the hypotenuse.

Pythagorean Theorem equation

X² + Y² = V² V = √X² +Y²

Scalars

are quantities without direction. Scalar quantities may be the magnitude of vectors, like speed, distance, energy, pressure, mass, or may be dimensionless, like coefficients of friction.

How is a vector calculated from the product of two vectors?

cross product: A × B = |A||B|sinθ

cos 45°

√2/2 or 1/√2

sin 45°

√2/2 or 1/√2

tan 60°

√3

cos 30°

√3/2

sin 60°

√3/2

Cross product equation

A × B = |A||B|sinθ

True or False: If C = A x B, where A is directed toward the right side of the page and B is directed to the top of the page, then C is directed midway between A and B at a 45° angle.

False. This would be true of an addition problem in which both vectors have equal magnitude, but it is never true for vector multiplication. To find the direction of C, we must use the right-hand rule. If the thumb points in the direction of A, and the fingers point in the direction of B, then our palm, C, points out of the page.

When calculating the sum of vectors A and B (A+B) we put the tail of B at the tip of A. What would be the effect of reversing this order (B+A)?

Vector addition, unlike vector multiplication, is a commutative function. The resultant of A + B is the same as B + A, so there would be no difference between the two resultants.

Multiplying a vector by a scalar

changes the magnitude and may reverse the direction.

How is a scalar calculated from the product of two vectors?

dot product: A ° B = |A||B|cosθ

Vector subtraction

is accomplished by changing the direction of the subtracted vector and then following the procedures for vector addition.

Vector addition

may be accomplished using the tip-to-tail method or by breaking a vector into its components and using the Pythagorean theorem.

Multiplying two vectors using the dot product

results in a scalar quantity, like work. The dot product is the product of the vectors' magnitudes and the cosine of the angle between them.

Multiplying two vectors using the cross product

results in a vector quantity. The cross product is the product of the vectors' magnitudes and the sine of the angle between them. The right-hand rule is used to determine the resultant vector's direction. It is not commutative.

Angle of the resultant vector from component vectors

θ = tan⁻¹ (Y/X)

What are the magnitudes and directions of the resultant vectors from the following cross products: C = A x B and D = B x A ? A: X = -3 N, Y = 0 B: X = 0, Y = +4 m

A × B = |A||B|sinθ A × B = |-3||+4|sin(90) = 12 x 1 = 12 N m C is 12 N m into the page, and D is 12 N m out of the page.

tan 30°

√3/3 or 1/√3

When calculating the difference of vectors A and B (A - B) we invert B and put the tail of this new vector at the tip of A. What would be the effect of reversing this order (B-A)?

Vector subtraction, like vector multiplication, is not a commutative function. The resultant of A - B has the same magnitude as B - A, but is oriented in the opposite direction.