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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

7. Vectors

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Exercise 10 Page 687

Review vector operations. Begin by substituting the vectors in the given expression.

Result: ⟨ 2, - 7 ⟩
Graph:

Practice makes perfect

We want to find the difference of the given vectors and then check our answer graphically. Let's do those things one at a time.

Difference of the Vectors

Before we begin, let's review vector operations.

Vector Addition	⟨ a,b ⟩ + ⟨ c,d ⟩ = ⟨ a+c,b+d ⟩
Vector Subtraction	⟨ a,b ⟩ - ⟨ c,d ⟩ = ⟨ a-c,b-d ⟩
Vector Multiplication	k ⟨ a,b ⟩ = ⟨ ka,kb ⟩

To simplify the given expression, we will begin by substituting the component form of the given vectors, a = ⟨ -4, 1 ⟩ and b = ⟨ -1,-3 ⟩, into the expression. 2b-a ⇓ 2 ⟨ -1,-3 ⟩ -⟨ -4, 1 ⟩ Now we can simplify the expression by first multiplying by the scalar and then subtracting the remaining terms.

2 ⟨ -1,-3 ⟩ -⟨ -4, 1 ⟩

▼

Simplify

Distribute the scalar

⟨ 2(-1),2(-3) ⟩ - ⟨ -4, 1 ⟩

a(- b)=- a * b

⟨ -2,-6 ⟩ - ⟨ -4, 1 ⟩

⟨ a,b ⟩ -⟨ c,d⟩ =⟨ a-c,b-d⟩

⟨ -2- (-4),-6-1 ⟩

a-(- b)=a+b

⟨ -2+4,- 6-1 ⟩

Add and subtract terms

⟨ 2, - 7 ⟩

Checking the Answer

Now, to check our answer, we will use the Parallelogram Method. To do so, we will start by graphing 2b and -a. 2b = & 2⟨ - 1,- 3 ⟩ = ⟨ - 2, - 6 ⟩ - a = & - ⟨ - 4, 1 ⟩ = ⟨ 4, - 1 ⟩ The vectors are given in the component form ⟨ x,y ⟩, which describes the vector in terms of its horizontal component x and vertical component y. Let's graph the vectors!

To complete the parallelogram we will draw the other two sides of the parallelogram. One side will have its initial point at the terminal point of - a and the length of 2b and the other will start at the terminal point of 2b and have the length of -a. Remember that the opposite sides of the parallelogram are parallel.

Now, we can draw the diagonal of the obtained parallelogram starting at the origin.

The diagonal represents the difference of 2b and a. Finally, let's consider the horizontal and vertical components of the obtained vector.

As we can notice, the vector 2b-a, has the component form ⟨ 2, - 7 ⟩. Therefore, our answer is correct.

Subchapter links

Exercises p.687-690