Problem 34 - Precalculus with Limits

Question

In Exercises 31-38, find (a) \(\small{\mathbf{u}} + \small{\mathbf{v}}\), (b) \(\small{\mathbf{u}} - \small{\mathbf{v}}\), and (c) \(\small{2\mathbf{u}} - \small{3\mathbf{v}}\), Then sketch each resultant vector. \(\mathbf{u} = \langle 0, 0 \rangle\), \(\mathbf{v} = \langle 2, 1 \rangle\)

Step-by-Step Solution

Answer

The results of the vector operations are (a) \(\mathbf{u} + \mathbf{v} = \langle 2, 1 \rangle\), (b) \(\mathbf{u} - \mathbf{v} = \langle -2, -1 \rangle\), and (c) \(\small{2\mathbf{u}} - \small{3\mathbf{v}} = \langle -6, -3 \rangle\).

Step 1: Vector Addition

To find the sum of the vectors \(\mathbf{u}\) and \(\mathbf{v}\), you add the corresponding components from each vector. Since \(\mathbf{u} = \langle 0, 0 \rangle\) and \(\mathbf{v} = \langle 2, 1 \rangle\), the sum is \(\mathbf{u} + \mathbf{v} = \langle 0+2, 0+1 \rangle = \langle 2, 1 \rangle\).

Step 2: Vector Subtraction

To find the difference of the vectors \(\mathbf{u}\) and \(\mathbf{v}\), you subtract the corresponding components from each vector. So, \(\mathbf{u} - \mathbf{v} = \langle 0-2, 0-1 \rangle = \langle -2, -1 \rangle\).

Step 3: Vector Scaling

This part involves scaling the vectors \(\mathbf{u}\) and \(\mathbf{v}\) by factors of 2 and -3 respectively, and then subtracting the results. So, \(\small{2\mathbf{u}} - \small{3\mathbf{v}} = 2\langle 0, 0 \rangle - 3\langle 2, 1 \rangle = \langle 0, 0 \rangle - \langle 6, 3 \rangle = \langle -6, -3 \rangle\).

Step 4: Sketching the vectors

The given vectors and their resultant vectors can be sketched on a 2-dimensional grid. The origin (0,0) represents vector \(\mathbf{u}\), point (2,1) is \(\mathbf{v}\), \(\mathbf{u} + \mathbf{v}\) ends at (2,1), \(\mathbf{u} - \mathbf{v}\) points to (-2,-1) and \(\small{2\mathbf{u}} - \small{3\mathbf{v}}\) ends at (-6,-3).

Key Concepts

Vector Addition

Vector addition in precalculus is a fundamental operation which combines two or more vectors into a single resultant vector. This process involves adding the corresponding components of each vector together. For example, if you have two vectors, \(\mathbf{u} = \langle a, b \rangle\) and \(\mathbf{v} = \langle c, d \rangle\), their sum is found by adding the 'x' components (a and c) and the 'y' components (b and d). Thus, the resultant vector \(\mathbf{u} + \mathbf{v}\) would be \(\langle a+c, b+d \rangle\).

It is helpful to visually represent vector addition on a graph by placing the tail of one vector at the head of the other and then drawing a vector from the origin to the head of the second vector. This visual representation is known as the 'tip-to-tail' method and can provide a clear geometric interpretation of the vector addition process.

Vector Subtraction

Vector subtraction, on the other hand, is the inverse operation to vector addition. It involves finding a vector that represents the difference between two given vectors. To subtract vector \(\mathbf{v}\) from vector \(\mathbf{u}\), you subtract the components of \(\mathbf{v}\) from the corresponding components of \(\mathbf{u}\), which results in the vector \(\mathbf{u} - \mathbf{v} = \langle a-c, b-d \rangle\).

In graphical terms, subtracting \(\mathbf{v}\) from \(\mathbf{u}\) can be thought of as adding the 'negative' of \(\mathbf{v}\). The 'negative' of a vector is another vector with the same magnitude but in the opposite direction. Thus, vector subtraction can be visualized using the same 'tip-to-tail' method as vector addition, but with the second vector reversed direction.

Vector Scaling

Vector scaling, also known as scalar multiplication, involves changing the magnitude of a vector without altering its direction. In the context of precalculus, scalar multiplication is performed by multiplying each component of a vector by a scalar (a real number). For instance, if you have a vector \(\mathbf{u} = \langle a, b \rangle\) and a scalar \(k\), scaling \(\mathbf{u}\) by \(k\) results in \(k\mathbf{u} = \langle ka, kb \rangle\).

This operation stretches or compresses the vector by a factor of \(k\), depending on whether \(k\) is greater or less than one, respectively. It's important to note that if \(k\) is negative, the direction of the vector is reversed as well as scaled. This concept is essential in many applications, such as physics, where vectors are used to represent quantities like force and velocity.

Vector Sketching

Vector sketching is the visual representation of vectors in a coordinate system, typically on a two-dimensional graph for vectors in \(\mathbb{R}^2\) or a three-dimensional space for vectors in \(\mathbb{R}^3\). To sketch a vector, you plot its 'tail' at the origin of the coordinate system and its 'head' at the point indicated by its components. For example, the vector \(\mathbf{v} = \langle x, y \rangle\) would be plotted with its tail at the origin (0,0) and its head at the point (x,y).

Sketching vectors can aid in understanding operations like addition, subtraction, and scaling by giving a geometric perspective on these processes. When sketching the result of vector operations, it's helpful to use different colors or styles for each vector to distinguish between them. Additionally, including the resultant vector in the sketch provides an immediate visual interpretation of the operation's outcome.