(1.2) Kinematics and Dynamics: Vectors and Scalars

Multiplying Vectors by Scalars. 1) With regard to magnitude and direction, what will happen when you multiply a vector by a scalar? 2) Explain how to find magnitude and direction of the product vector using the equation B=nA 3) For example, if vector A is multiplied by the scalar +3, the new vector B is ________ times as long and points in the _________ direction.

1) The vectors magnitude will change and its direction will either be parallel or antiparallel to it's original direction. 2) In this equation B is the product vector of n (the scalar) and A (the original vector). To find the magnitude, you multiply the magnitude of A by the absolute value of n. To determine the direction, you look at the sign on n - if n is a positive number, B and A are the same direction. If its negative, they are in the opposite direction. 3) three, same

1) If given the value of a known vector, how do you break it into its perpendicular components? 2) Say you are given a vector of unknown value, but you have the value of it's perpendicular (x,y) components. How to you calculate the value of the vector? 3) Say you have three vectors, all of which you have the component values of x,y, and V. How do you use this information calculate the resultant of the three vectors?

1) To break a vector into it's perpendicular components (x-y components), you draw a right triangle with vector V as the hypotenuse and using trigonometry relationships (SOH-CAH-TOA) based on the the angle between V and the x component (Theta θ) to find the values of the x and y components (the other two sides of the right triangle). 2) Conversely, if we know the value of the vector's x and y components, we can use the Pythagorean theorem (A^2 + B^2 = C^2) to calculate the missing side of the triangle (which is the vector V). 3) You sum the x components and y components of each vector to get Rx and Ry values - these are the x and y values of the resultant vector, respectively. Since you have the values of Rx and Ry at this point, you can use the Pythagorean Theorem to calculate the value of the Resultant Vector (Rv) using same steps #2 above.

Intro to Vectors and Scalars. 1) Differentiate between a vector and a scalar. 2) Give some examples of both vector quantities and scalar quantities. 3) How are vectors and scalars represented when drawn? What about in notation?

1) Vectors are numbers that have magnitude and direction. Scalars are numbers that have magnitude only and no direction. 2) Vectors = displacement, velocity, acceleration, and force. Scalars = distance, speed, energy, pressure, and mass. 3) Vectors are represented by arrows, where the direction of the arrow indicates the direction of the vector and the length of the arrow is usually proportional to the magnitude of the vector quantity. When notated, the straight line path of a vector can be identified as a bolded A (which can have an arrow drawn over the top), and the magnitude of displacement between the two positions can be represented as the absolute value of that. Scalar quantities as well as the magnitude of a vector are usually represented with italic type. For example, the distance between two points could be represented by an italicized d.

Vector Addition. 1) The sum or difference between two or more vectors is called the __________ of the vectors. 2) What are the two main ways to sum two vectors?

1) resultant 2) Two main ways are the "Tip to Tail" method and the "Components" method.

*Multiplying Vectors by Other Vectors. 1) In some circumstances, we want to be able to use two vector quantities to generate a third vector or scalar by multiplication. To generate a __________ quantity, like work, we multiply the magnitudes (the absolute values) of the two vectors of interest and the __________ of the angle between the two vectors. In vector calculus, this is called the ____________. In contrast, to generate a third ___________ quantity, like torque, we multiply the magnitudes of the two vectors of interest and the __________ of the angle between the two vectors. In vector calculus, this is called the __________. 2) When multiplying two vectors to generate a third vector, we need to determine both it's magnitude and direction. Direction of this plane is determined using the ___________. Explain how this works. 3) T/F - The resultant of a cross product will always be parallel to the plane created to by two vectors.

1) scalar, cosine, dot-product; vector, sine, cross product. 2) using the right hand rule; See picture; In the multiplication of either vector AxB or BxA, the the thumb will be oriented as the first vector. That is, A (for AxB) or B (for BxA). 3) False - always will be perpendicular. this means it will usually be going into or out of the page (think right hand rule...see picture).

Vector Addition: Finding the X and Y components. 1) In vector addition, theta (θ) references the angle between the ___________ and the _______________. 2) The trigonometric equations for calculating the X and Y components using theta are _____________ and ____________, respectively 3) The equation for calculating the angle of the resultant vector (theta) is ______________.

1) vector and the x-component 2) X=Vcosθ and Y=Vsinθ, respectively 3) Theta = inverse tan (Y/X)

Provide the equations for the following: 1. Determination of direction from component vectors 2. Dot product 3. Cross product

1. see image (equation 1.3) 2. see image (equation 1.4) 3. see image (equation 1.5)

Vector Subtraction. How an subtracting one vector from another be accomplished?

Essentially, by turning it into a vector addition problem. You do this by adding a vector with equal magnitude, but opposite direction, to the first vector (A-B) becomes A + (-B). From here on, you just add tip to tail or break it into its components and perform vector addition to find the resultant.

Practice. 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 degree 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, you 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.

Explain how to calculate the resultant of two vectors using the "Tip to Tail" method.

In the tip-to-tail method, you join the tip of vector A to the tail of vector B, making the sum (resultant) of the two vectors A+B.

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

The scalar is calculated from two vectors by using the dot product: AxB = |A| x |B| x CosTheta. A Vector is calculated by using the cross product: AxB = |A| x |B| SinTheta

Practice. When calculating the sum of the 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)

There would be no difference in the resultants because vector addition is a communitive function, unlike vector multiplication.

T/F: If one wishes to use the "Tip to Tail Method" for vector addition, the lengths of arrows must be proportional to the magnitudes of the vectors.

True

Practice. 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 effects of referring this order (B-A)?

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