Math 120 - Cooley Trigonometry OCC

Section 5.4 – Vectors Vector addition is commutative. That is, if v and w are any two vectors, then

v + w = w + v

Vector addition is also associative. That is, if u, v, and w are vectors, then

u + (v + w) = (u + v) + w

The zero vector 0 has the property that v + 0 = 0 + v = v for any vector v. Definition

If 𝛼 is a scalar and v is a vector, the scalar multiple 𝛼v is defined as follows:

1. If 𝛼 > 0, 𝛼v is the vector whose magnitude is 𝛼 times the magnitude of v and whose direction is the

same as v.

2. If 𝛼 < 0, 𝛼v is the vector whose magnitude is |𝛼| times the magnitude of v and whose direction is

opposite that of v.

3. If 𝛼 = 0 or if v = 0, then 𝛼v = 0.

Note that when a v or 0 is written in bold that this represents “vector v” or “vector 0”, respectively. Theorem – Properties of ||v||

If v is a vector and if 𝛼 is a scalar, then

(a) ||v|| ≥ 0 (b) ||v|| = 0 if and only if v = 0. (c) ||–v|| = ||v||

(d) || 𝛼v|| = |𝛼|||v||

1 Math 120 - Cooley Trigonometry OCC

Section 5.4 – Vectors Definition - A vector u for which ||u|| = 1 is called a unit vector. Definition - An algebraic vector v is represented as v = a, b where a and b are real numbers (scalars) called the components of the vector v. Theorem Suppose that v is a vector with initial point P= ( x, y ) , not necessarily the origin, and terminal point  1 1 1 P2= ( x 2, y 2 ) . If v = PP1 2 , then v is equal to the position vector

v = x2- x 1, y 2 - y 1

Theorem – Equality of Vectors Two vectors v and w are equal if and only their corresponding components are equal. That is,

If v = a1, b 1 and w = a2, b 2

then v = w if and only if a1 = a2 and b1 = b2. Definition

Let v = a1i + b1j = a1, b 1 and w = a2i + b2j = a2, b 2 be two vectors, and let α be a scalar. Then

v + w = (a1 + a2)i + (b1 + b2)j = a1+ a 2, b 1 + b 2

v – w = (a1 – a2)i + (b1 – b2)j = a1- a 2, b 1 - b 2

αv = (αa1)i + (αb1)j = aa1, a b 1

2 2 ||v|| = a1+ b 1 Theorem – Unit Vector in the Direction v For any nonzero vector v, the vector v u = ||v || is a unit vector that has the same direction as v. (Written in terms of v as v=|| v || u ).

 Exercises: The vector v has initial point P and terminal point Q. Write v in the form of a i + b j ; that is, find its position vector.

1) P = (0,0); Q = (–3, –5) 2) P = (–1,4); Q = (6, 2)

Section 5.4 – Vectors

 Exercises: Find || v || .

3) v = –5i + 12j 4) v = 6i + 2j

 Exercises: Find each quantity if v = 3 i – 5 j and w = –2 i + 3 j .

5) 3v – 2w 6) ||v + w|| 7) ||v|| + ||w||

Section 5.4 – Vectors  Exercises: Find the unit vector in the same direction as v .

8) v = –3j 9) v = 2i – j

 Exercises: Write the unit vector v in the form a i – b j , given its magnitude || v || and the angle 𝛼 it makes with

the positive x-axis.

10) ||v|| = 8, 𝛼 = 45° 11) ||v|| = 15, 𝛼 = 315°

 Exercises:

12) A man pushes a wheelbarrow up an incline of 20° with a force of 100 pounds. Express the force vector F in terms of i and j.

13) Two forces of magnitude 30 newtons (N) and 70 N act on an object at angles of 45° and 120° with the

positive x-axis, as shown in the figure. Find the direction and magnitude of the resultant force; that is, find F1 + F2.

Section 5.4 – Vectors

Section 5.4 – Vectors  Exercises:

14) Static Equilibrium. A weight of 800 pounds is suspended from two cables as shown in the figure. What are the tensions in the two cables?