MATH 1330 Vectors

Vectors in a plane Vectors have a magnitude (size or strength) The arrow at the terminal point does not mean that the vector continues forever in that direction. It is only to indicate direction. the terminal point is the ending point Vectors have a magnitude (size or strength) Vectors have a direction (slope or directional angle). The initial point is the starting point

Component Form of a Vector To determine the component form of a vector, v, with initial point P (a, b), and terminal point Q (c,d), you must subtract: terminal point – initial point. v = <c – a, d – b>.

Component Form of a Vector To determine the component form of a vector, v, with initial point P (a, b), and terminal point Q (c,d), you must subtract: terminal point – initial point. v = <c – a, d – b>. This is the vector translated so that the initial point is at (0,0).

Place the following vectors into component form: u: Initial Point: P (4, -2); Terminal Point (5, 1).

Place the following vectors into component form: u: Initial Point: P (4, -2); Terminal Point (5, 1). u = <5 - 4, 1 - (-2)> = <1, 3>

Place the following vectors into component form: u: Initial Point: P (4, -2); Terminal Point (5, 1). Notice, this is the coordinates of the “new” terminal point u = <5 - 4, 1 - (-2)> = <1, 3>

Place the following vectors into component form: u: Initial Point: P (4, -2); Terminal Point (5, 1). Also, notice that they will both have the magnitude and direction. u = <5 - 4, 1 - (-2)> = <1, 3>

Find the component form of: v: Initial Point: P (0, 4); Terminal Point (9, -3). w: Initial Point: P (-2, 5); Terminal Point (7, 2).

Finding Magnitude and Directional Angle: If you needed to calculate the distance between the terminal and initial points of a vector, what formula can you use?

Finding Magnitude and Directional Angle: If you needed to calculate the distance between the terminal and initial points of a vector, what formula can you use? The Distance Formula: 𝑑= 𝑥 2 − 𝑥 1 2 + 𝑦 2 − 𝑦 1 2

Finding Magnitude : If you needed to calculate the distance between the terminal and initial points of a vector, what formula can you use? The Distance Formula: 𝑑= 𝑥 2 − 𝑥 1 2 + 𝑦 2 − 𝑦 1 2 If you had the vector in standard position, how would the formula simplify?

If v = <v1, v2>: 𝑑= 𝑣 1 −0 2 + 𝑣 2 −0 2 𝑣 = 𝑣 1 2 + 𝑣 2 2

If v = <v1, v2>: | 𝑣 |= 𝑣 1 2 + 𝑣 2 2 𝑣 = 𝑣 1 2 + 𝑣 2 2 This means the Magnitude of v

Finding Directional Angles If you needed to calculate the angle between the positive x-axis and a vector in standard position, how would you use this?

Finding Directional Angles If you needed to calculate the angle between the positive x-axis and a vector in standard position, how would you use this? How can the unit circle be used here? What trigonometric functions can be used?

v = <v1, v2>

Finding Directional Angles tan 𝜃 = 𝑣 2 𝑣 1

Finding Directional Angles tan 𝜃 = 𝑣 2 𝑣 1 Make sure you account for quadrant when you do this!

Writing Vectors Any vector can be defined by the following: v = <||v||cos θ, ||v||sin θ>

Determine the magnitude and direction of: <8, 6>, <-3, 5>

Determine the component form of the vector with magnitude of 5 and directional angle of 120o. Popper 20: Question 1: a. 5 2 , 5 2 b. −5 2 , 5 3 2 c. −5 3 2 , 5 2 d. 5 2 , 5 3 2

Vector Operations: <a, b> + <c, d> = <a + c, b + d> <a, b> - <c, d> = <a - c, b - d> k <a, b> = <ka, kb>

Resultant Force The Chair Example!

Resultant Force: Popper 20, Continued Two forces are acting on an object. The first has a magnitude of 10 and a direction of 30o. The other has a magnitude of 5 and a direction of 45o. Determine the magnitude and direction of their Resultant Force. Question 2: First Vector: a. 5 3 ,5 b. 5 3 ,5 3 c. 5,5 3 d. 10 3 ,5

Resultant Force: Popper 20, Continued Two forces are acting on an object. The first has a magnitude of 10 and a direction of 30o. The other has a magnitude of 5 and a direction of 45o. Determine the magnitude and direction of their Resultant Force. Question 3: Second Vector: a. 5 2 ,−5 b. 5 2 2 ,5 2 c. 5 2 2 , 5 2 2 d. 5 2 2 , 2 2

Resultant Force: Popper 20, Continued Two forces are acting on an object. The first has a magnitude of 10 and a direction of 30o. The other has a magnitude of 5 and a direction of 45o. Determine the magnitude and direction of their Resultant Force. Question 4: Resultant Vector (as decimals): a. 12.2,2.5 b. 1.5,8.5 c. 12.2,8.5 d. 11.3,5.1

Resultant Force: Popper 20, Continued Two forces are acting on an object. The first has a magnitude of 10 and a direction of 30o. The other has a magnitude of 5 and a direction of 45o. Determine the magnitude and direction of their Resultant Force. Question 5: Magnitude of Resultant a. 14.9 b. 4.5 c. 20.7 d. 1.9

Resultant Force: Popper 20, Continued Two forces are acting on an object. The first has a magnitude of 10 and a direction of 30o. The other has a magnitude of 5 and a direction of 45o. Determine the magnitude and direction of their Resultant Force. Question 6: Direction of Resultant a. 55.1o b. 34.9o c. 44.2o d. 45.8o

Look at this situation graphically (parallelogram) or analytically (operations on vectors).

Unit Vectors To calculate a unit vector, u, in the direction of v you must calculate: u = (1/||v||)<v1, v2>.

Find the unit vector in the direction of the following: <3, 5> <1, 8>.

Linear Combination Form: Standard Unit Vectors: i = <1,0> j = <0,1>

Linear Combination Form: Standard Unit Vectors: i = <1,0> j = <0,1> If v = <v1, v2> = v1 <1, 0> + v2 <0, 1> = v1 i + v2 j

So convert the following w = <3, -5> into linear combination form.

The Dot Product of Two Vectors Vocabulary: Angle between vectors: The smallest angles between two vectors in standard position Orthogonal Vectors: Vectors that are at right angles.

Calculating the dot product of two vectors Consider u = <a, b> and v = <c, d> u · v = ac + bd

Determine value of <7, 5> · <9, -1> Determine value of <6, 1> · <-5, 3>

Angles between Vectors To find the angle between vectors, you must calculate: cos 𝜃 = 𝑢∙𝑣 𝑢 ∙ 𝑣

Determine the angle between the vectors: u = <9, 3>; and v = <4, 8>.

Determine the angle between the vectors: u = <0, 4>; and v = <3, 9>.

Give a possible vector that would be at a right angle to <7, -2>?

Give a possible vector that would be at a right angle to <7, -2>? What general rule can you use to determine orthogonal vectors?