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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

7. Vectors

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Exercise 45 Page 689

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

⟨ 26, 5 ⟩

Practice makes perfect

Before we begin, let's review vector operations. Let ⟨ a,b ⟩ and ⟨ c,d⟩ represent vectors and k be a scalar.

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 ⟩

Now, to simplify the given expression, we will begin by substituting the component form of the given vectors, f = ⟨ -4, -2 ⟩, g = ⟨ 6,1 ⟩, and h = ⟨ 2, -3 ⟩, into our expression. Let's do it! 2g-3f+h ⇓ 2 ⟨ 6,1 ⟩ -3 ⟨ -4, -2 ⟩ + ⟨ 2, -3 ⟩ Next, we can simplify the expression. First we will perform multiplication by the scalar and then we will add and subtract the remaining vectors.

2 ⟨ 6,1 ⟩ -3 ⟨ -4, -2 ⟩ + ⟨ 2, -3 ⟩

▼

Simplify

Distribute the scalar

⟨ 2 * 6,2 * 1 ⟩ - ⟨ 3(-4), 3(-2) ⟩ + ⟨ 2, -3 ⟩

Multiply

⟨ 12,2 ⟩ - ⟨ -12, -6 ⟩ + ⟨ 2, -3 ⟩

⟨ a,b ⟩ -⟨ c,d⟩ =⟨ a-c,b-d⟩

⟨ 12- (-12),2-(-6) ⟩ + ⟨ 2, -3 ⟩

a-(- b)=a+b

⟨ 24,8 ⟩ + ⟨ 2, -3 ⟩

⟨ a,b⟩+⟨ c,d⟩=⟨ a+c,b+d⟩

⟨ 24+2,8+(-3) ⟩

Add and subtract terms

⟨ 26, 5 ⟩

Extra

Component Form of a Vector

Let's recall that we use the component form of a vector to describe the vector in terms of its horizontal component x and vertical component y. ⟨ x,y ⟩ In other words, the component form of a vector tells us how many units, either in vertical or horizontal directions, we should move from the initial point to end in the terminal point.

Subchapter links

Exercises p.687-690