Math 1330 Vectors Vectors on a Plane: Vocabulary: Vector v in the plane, Standard position, Zero vector, Unit vector, Standard unit vectors, Direction angle Vectors have a magnitude (size or strength) and a direction (slope or directional angle). The initial point is the starting point, and the terminal point is the ending point. 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> Example 1: Place the following vectors into component form: u: Initial Point: P (4, ‐2); Terminal Point (5, 1). v: Initial Point: P (0, 4); Terminal Point (9, ‐3). w: Initial Point: P (‐2, 5); Terminal Point (7, 2). pg. 1 Math 1330 Vectors 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? Example 2: Given the initial point 3, 3 and terminal point 0,3 , find the magnitude. If you had the vector in standard position, how would the formula simplify? Example 3: Given the initial point 0,0 and terminal point 3,6 , find the magnitude. ‖ ‖
Since the vectors in examples 2 and 3 have the same magnitude now we can check the direction to see if these vectors are equal. One way is to check the slope of the lines which they lie. Both u and v are directed the toward the upper right hand on lines that have the same slope so the conclusion is they have the same direction: u =v have the same magnitude and direction. Scalar Multiplication: If k is a real number and v a vector , kv is called the scalar multiple of a vector v. The magnitude of kv is given by |k| ||v||. The vector kv has a direction.  the same direction of v if k > 0.  opposite direction of v if k < 0. pg. 2 Math 1330 Vectors The geometric method for adding vectors looks like: v u + v u The sum of u + v is called the resultant vector. The difference of two vectors v  u looks like:  u v v  u u u Vectors in the rectangular coordinate system: Vector v, with initial point v = ,
and the terminal point i + ,
is equal to the position vector j Example 4: Express a vector with initial point 3, 1 and terminal point vectors.  the horizontal component of v is:  the vertical component of v is:  find the magnitude of v: pg. 3 2, 5 linear combination of standard Math 1330 Vectors Example 5: If v = 5 i + 4 j and w = 6 i  9 j a. v + w b. v  w c. 4v  2w Finding a unit vector that has the same direction: || ||
Example 6: Find the unit vector that has the same direction: v = 5 i  12 j 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? tan
Any vector can be defined by the following: v = <||v||cos θ, ||v||sin θ> pg. 4 Math 1330 Vectors Example 7: Find the direction angle that this vector makes with the positive x axis. a. v = i + √3 j b. u = 〈16, 4〉 c. Given vector u = 〈1, x〉 and has magnitude and that its directional angle with the positive x‐axis is acute, what is the value of x? Example 8: Find a vector that has a magnitude of 20 and that has a direction of . Example 9: A vector has a magnitude 12 and it has a direction angle of 150 with the positive x‐axis. Find the horizontal component of this vector. pg. 5 Math 1330 Vectors The Dot Product of Two Vectors Vocabulary: Angle between vectors, Orthogonal Vectors Calculating the dot product of two vectors Consider u = <a, b> and v = <c, d> u · v = ac + bd Example 10: Find the dot product. v = 5 i  2 j and u = 3 i 4 j A. u · v B. v u C. vv Properties of Dot Products: Angles between Vectors: To find the angle between vectors, you must calculate: cos
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Math 1330 Vectors Example 11: Let u = 2 i  j and v =  4i +2j. Find the angle between these vectors. Determining whether vectors are orthogonal. Two non‐zero vector v and w are orthogonal if and only if: v  w = 0 Example 12: Are the vectors v = 2 i +3 j and w = 6 i  4 j orthogonal? Example 13: Are the vectors v = 2 i  2 j and w =  i + j orthogonal? Example 14: Let v = 2 i + 8 j . Find a vector that is orthogonal to this vector. Example 15: Find a value for b so that u = < 15, 3> and v = < 4, b > are orthogonal. pg. 7 Math 1330 Popper 22
1. Express a vector with initial point 4, 1 and terminal point
combination of standard vectors.
A. v =  6 i + 4 j
2. Find the unit vector that has the same direction: v = 3 i  4 j
i A.
3. Given: v = 3 i + 7j , find 5v
A.  15 I +35 j
4. Find the dot product: v = 3 i + j and w = i + 3 j
A.
6
5. Find if vectors: v = i + j and w = i  j are orthogonal.
A. Yes
2, 3 linear