Exercise 36 Page 688

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

Sum 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 = ⟨ -3, -5 ⟩ and c = ⟨ 3, -1 ⟩, into the expression. c+a ⇓ ⟨ 3, -1 ⟩ + ⟨ -3, -5 ⟩ Now we can simplify the expression by adding the vectors.

⟨ 3, -1 ⟩ + ⟨ -3, -5 ⟩

▼

Simplify

⟨ a,b⟩+⟨ c,d⟩=⟨ a+c,b+d⟩

⟨ 3+(-3),-1+(-5) ⟩

AddNeg

a+(- b)=a-b

⟨ 3-3,-1-5 ⟩

SubTerms

Subtract terms

⟨ 0,-6 ⟩

Checking the Answer

Now, to check our answer, we will use the Parallelogram Method. To do so, we will start by graphing a=⟨ - 3, - 5 ⟩ and c=⟨ 3, - 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 c and the other will start at the terminal point of c 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 sum of c and a. As we can notice, the vector c+a, has the component form ⟨ 0, - 6 ⟩. Therefore, our answer is correct.

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Sum of the Vectors

Checking the Answer