Vectors Slide 2 / 36 Scalar versus Vector A scalar has only a - PowerPoint PPT Presentation

Slide 1 / 36 Vectors

Slide 2 / 36 Scalar versus Vector A scalar has only a physical quantity such as mass, speed, and time. A vector has both a magnitude and a direction associated with it, such as velocity and acceleration. A vector is denoted by an arrow above the variable,

Slide 3 / 36 Is this a vector or a scalar? 1 Scalar Time Scalar Speed Velocity Vector Scalar Distance Vector Displacement

Slide 4 / 36 Which of the following is a true statement? 2 It is possible to add a scalar quantity to a vector. A The magnitude of a vector can be zero even B though one of its components is not zero. The sum of the magnitude of two C unequal vectors can be zero. Rotating a vector about an axis passing through D the tip of the vector does not change the vector. Vectors must be added geometrically. E

Slide 5 / 36 Drawing A Vector A vector is always drawn with an arrow at the tip indicating the direction, and the length of the line determines the magnitude. Remember displacement is the distance away from your initial position, it does not account for the actual distance you moved

Slide 6 / 36 Determining magnitude and direction anti-parallel All of these vectors have the same magnitude, but vector B runs anti-parallel therefore it is denoted negative A.

Slide 7 / 36 Vector Addition

Slide 8 / 36 Vector Addition Methods Tail to Tip Method

Slide 9 / 36 Vector Addition Methods Parallelogram Method Place the tails of each vector against one another. Finish drawing the parallelogram with dashed lines and draw a diagonal line from the tails to the other end of the parallelogram to find the vector sum.

Slide 10 / 36 3 If a car under goes a displacement of 3 km North and another of 4 km to the East what is the net displacement? A 5√2 km 4 km B 5 km 3 km x C 4√3 km D 7 km E 6 km

Slide 11 / 36 4 If a car under goes a displacement of 3 km North and another of 4 km to the East what is the total distance traveled? A 5√2 km 4 km B 7 km 3 km x C 5 km D 4 km E 3 km

Slide 12 / 36 5 Solve for θ 45 o A 4 km 75 o B 53 o C 3 km x 37 o D θ 25 o E

Slide 13 / 36 Vector Components v v y θ v x A vector that makes an angle with the axis has both a horizontal and vertical component of velocity. θ is measured starting at the x axis and rotating in the direction of the y-axis.

Slide 14 / 36 Multiple Vectors When dealing with multiple vectors you can just add the components in order to attain the components of the vector sum. v x v y v y v x v x v y v y v x

Slide 15 / 36 6 The components of vector A are given as follows: The magnitude of A is closest to: A 4.2 B 8.4 11.8 C 18.9 D E 70.9

Slide 16 / 36 7 The components of vectors and are given as follows: Solve for the magnitude of A 5 B √17 C 17 10 D E 8

Slide 17 / 36 8 The components of vector A are given as follows: The angle measured counter-clockwise from the x-axis to vector A, in degrees, is closest to: 339 o A 200 o B 122 o C 21 o D 159 o E

Slide 18 / 36 9 The components of vector A and B are given as follows: The magnitude of B - A, is closest to: A 10.17 B 4.92 2.8 C 9.7 D E 25

Slide 19 / 36 10 The magnitude of B is 5.2. Vector B lies in the 4th quadrant and forms a 30 o with the x-axis. The components of B x and B y are: A B C D

Slide 20 / 36 11 The magnitude of vector A is equal to vector B plus vector C. What is the value of vector A? A 2.59 y B -1.78 5.3 C -3.42 D 1.63 x 30 O E -2.5 45 O 6

Slide 21 / 36 12 Vectors A and B are shown. Vector C is given by C = A + B. In the figure above, the magnitude of C is closest to: A 7.5 B 3.9 C 5.2 30 o o 60 D 9.3 E 2.6

Slide 22 / 36 Unit Vectors Unit vectors have no units and a magnitude of 1. Unit Vectors describe a direction in space. indicates the x direction indicates the y direction indicates the z direction Any given Vector can be presented in terms of unit vectors:

Slide 23 / 36 Unit Vectors When two vectors A and B are presented in terms of their components, we can express the vector sum R using unit vectors:

Slide 24 / 36 13 What is the magnitude of the sum of the following vectors? A 9.3 B 12.3 5.1 C 10.7 D E 3

Slide 25 / 36 Products of Vectors Scalar Product also known as Dot Product yields a scalar quantity value can be positive, zero, or negative depending on θ. θ ranges from 0 to 180 degrees. = = = = = =

Slide 26 / 36 14 In the figure, find the scalar product of vectors B and C, A 0 B 17 4 24 C 7 -17 D 65 o 45 o 6 E -24

Slide 27 / 36 15 In the figure, find the scalar product of vectors A and C, A 0 B 14 4 42 C 7 -14 D 6 o o 65 45 E -42

Slide 28 / 36 Products of Vectors Vector Product also known as the cross product yields another vector. = = = = - = = - = = - =

Slide 29 / 36 16 In the figure, find the vector product of vectors A and B. 12 A B 30 4 C 25 7 20 D 65 o 45 o 6 E 10

Slide 30 / 36 17 Two vectors are give as follows: Solve for A B C D E

Slide 31 / 36 18 Two vectors are give as follows: Solve for A 2 B -4 7 C 5 D E -12

Slide 32 / 36 19 Which of the following is an accurate statement? If the vectors A and B are each rotated through the same angle about the same axis, the product A will be unchanged. If the vectors A and B are each rotated through the same angle about the same axis, the B product A x B will be unchanged If a vector A is rotated about an axis parallel to C vector B, the product will be changed. When a scalar quantity is added to a vector, the result is a vector of larger-magnitude than the D original vector.

Slide 33 / 36 20 Solve for the angle between vector and 97.93 o A 277.93 o B 57 o C 84.73 o D 124.38 o E

Slide 34 / 36 21 Two vectors are given: The angle between vectors A and B, in degrees, is: 117 o A 76 o B 150 o C 29 o D 161 o E

Slide 35 / 36 Two vectors are given: 22 Solve for the magnitude of 33 A B 29 25 C 21 D E 17

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