Problem 22 - Precalculus

Question

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$(\mathbf{a}+\mathbf{b})+\mathbf{c}$$

Step-by-Step Solution

Answer

The resultant vector is \(\langle 13, 6 \rangle\).

Step 1: Vector Addition of a and b

First, let's find the resultant vector of adding \( \mathbf{a} \) and \( \mathbf{b} \). The addition of two vectors \( \mathbf{a} = \langle 2, 3 \rangle \) and \( \mathbf{b} = \langle 5, 4 \rangle \) is performed by adding their respective components:\[\mathbf{a} + \mathbf{b} = \langle 2 + 5, 3 + 4 \rangle = \langle 7, 7 \rangle\]

Step 2: Add the Resultant to Vector c

Now that we have \( \mathbf{a} + \mathbf{b} = \langle 7, 7 \rangle \), add this vector to \( \mathbf{c} = \langle 6, -1 \rangle \). Again, add the corresponding components of the vectors:\[(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \langle 7, 7 \rangle + \langle 6, -1 \rangle = \langle 7 + 6, 7 + (-1) \rangle = \langle 13, 6 \rangle\]

Step 3: Conclusion

The result of the vector operation \((\mathbf{a} + \mathbf{b}) + \mathbf{c}\) is a new vector \(\langle 13, 6 \rangle\).

Key Concepts

Vectors

Vectors are mathematical objects used to represent quantities that have both magnitude and direction. In everyday life, common examples of vectors include forces, velocities, and displacements. Each vector is defined by two or more components. These components are often written as an ordered pair or triple, representing different dimensions.
For example, in two-dimensional space, a vector

• can be written as \( \mathbf{a} = \langle x, y \rangle \) where \( x \) and \( y \) are the components
• represents movement or direction on a plane.

Vectors are fundamental in physics and engineering, as they simplify the representation of physical phenomena. They provide a handy way to perform complex calculations like forces acting at various angles in physics.

Vector Components

The components of a vector are its projections along the coordinate axes. Let's take the vector \( \mathbf{a} = \langle 2, 3 \rangle \) as an example.
Here, the numbers 2 and 3 are its components.

• The first component (2) describes the horizontal change.
• The second component (3) describes the vertical change.

The components tell us where the vector is heading and how far it will go in that direction.
When adding vectors, each component of one vector is simply added to the corresponding component of the other vector. Think of it like combining steps in each direction. This way, you get a new path that correctly reflects both original paths combined.

Resultant Vector

A resultant vector is the vector result of adding two or more vectors together. It represents the combined effect of all those vectors.
For example, if a force of 5 N east and a force of 3 N north act on an object simultaneously, the resultant vector can be found using vector addition.
Let's see how this concept applies in our given problem:

• Start by adding vectors \( \mathbf{a} = \langle 2, 3 \rangle \) and \( \mathbf{b} = \langle 5, 4 \rangle \).
• Their resultant is \( \langle 7, 7 \rangle \), showing the combination of both forces.
• Then, add this result to vector \( \mathbf{c} = \langle 6, -1 \rangle \).
• This gives us the final resultant vector \( \langle 13, 6 \rangle \).

The resultant vector provides a single vector that has the same effect as the individual vectors acting together. It's a crucial concept for simplifying problems involving multiple vector quantities.