Problem 36 - Precalculus Mathematics for Calculus

Question

Find \(2 u,-3 v, u+v,\) and \(3 u-4 v\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v}\) $$\mathbf{u}=\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}-\mathbf{j}$$

Step-by-Step Solution

Answer

Find the vectors as follows: \(2\mathbf{u} = (2, 2)\), \(-3\mathbf{v} = (-3, 3)\), \(\mathbf{u} + \mathbf{v} = (2, 0)\), \(3\mathbf{u} - 4\mathbf{v} = (-1, 7)\).

Step 1: Understanding Vector Notation

Vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given in terms of their components along the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). Specifically, \(\mathbf{u} = \mathbf{i} + \mathbf{j}\) and \(\mathbf{v} = \mathbf{i} - \mathbf{j}\). This translates to components \(\mathbf{u} = (1, 1)\) and \(\mathbf{v} = (1, -1)\).

Step 2: Calculating 2u

The vector \(2\mathbf{u}\) is calculated by multiplying each component of \(\mathbf{u}\) by 2: \[2\mathbf{u} = 2(\mathbf{i} + \mathbf{j}) = 2\mathbf{i} + 2\mathbf{j} = (2, 2)\].

Step 3: Calculating -3v

The vector \(-3\mathbf{v}\) is calculated by multiplying each component of \(\mathbf{v}\) by -3: \[-3\mathbf{v} = -3(\mathbf{i} - \mathbf{j}) = -3\mathbf{i} + 3\mathbf{j} = (-3, 3)\].

Step 4: Calculating u+v

To find \(\mathbf{u} + \mathbf{v}\), add the corresponding components from both \(\mathbf{u}\) and \(\mathbf{v}\): \[\mathbf{u} + \mathbf{v} = (1 + 1, 1 - 1) = (2, 0)\].

Step 5: Calculating 3u - 4v

First calculate \(3\mathbf{u}\) and \(4\mathbf{v}\): \[3\mathbf{u} = 3(\mathbf{i} + \mathbf{j}) = 3\mathbf{i} + 3\mathbf{j} = (3, 3)\] and \[4\mathbf{v} = 4(\mathbf{i} - \mathbf{j}) = 4\mathbf{i} - 4\mathbf{j} = (4, -4)\]. Then subtract \(4\mathbf{v}\) from \(3\mathbf{u}\): \[3\mathbf{u} - 4\mathbf{v} = (3, 3) - (4, -4) = (3 - 4, 3 + 4) = (-1, 7)\].

Key Concepts

Vector Addition

Vector addition is a fundamental operation that combines two or more vectors to produce a new vector. It involves adding the components of each vector separately. For instance, suppose we have two vectors in two-dimensional space, \( \mathbf{u} \) and \( \mathbf{v} \), with \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \). Then their sum \( \mathbf{u} + \mathbf{v} \) is given by adding the corresponding components: \[ (u_1 + v_1, u_2 + v_2) \]. An example from our original exercise is adding \( \mathbf{u} = (1, 1) \) and \( \mathbf{v} = (1, -1) \). Their sum is \( (1 + 1, 1 - 1) = (2, 0) \).

• This sum represents a vector pointing in the positive x-direction, having zero y-component.
• Visualizing this on a graph shows that the vectors \( \mathbf{i} \) (x-axis) and \( \mathbf{j} \) (y-axis) help move the sum vector to its resulted destination in the plane.

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a single number, known as a scalar. If you have a vector \( \mathbf{u} = (u_1, u_2) \) and a scalar \( c \), the resulting vector after scalar multiplication, \( c\mathbf{u} \), is calculated by multiplying each component of the vector by the scalar: \[ c\mathbf{u} = (c \times u_1, c \times u_2) \]. In our exercise, we see this concept in practice with calculations like \( 2\mathbf{u} \) and \( -3\mathbf{v} \):

• For \( 2\mathbf{u} \), multiply each component of \( \mathbf{u} = (1, 1) \) by 2, resulting in \( (2, 2) \).
• For \( -3\mathbf{v} \), multiply each component of \( \mathbf{v} = (1, -1) \) by -3, resulting in \( (-3, 3) \).

The resulting vectors scale in magnitude but maintain or reverse direction depending on whether the scalar is positive or negative.

Vector Components

Vector components are key elements that define a vector with respect to a coordinate system. They express how much influence a vector has in each dimension, for example along the x-axis and y-axis. Consider a vector \( \mathbf{u} = \mathbf{i} + \mathbf{j} \). Here, \( \mathbf{i} \) and \( \mathbf{j} \) are standard unit vectors pointing along the axes:

• \( \mathbf{i} = (1, 0) \) represents a step of one unit along the x-axis.
• \( \mathbf{j} = (0, 1) \) represents a step of one unit along the y-axis.

The notation \( (1, 1) \) for \( \mathbf{u} \) implies a combination of both these unit vectors. This means \( \mathbf{u} \) moves one unit right and one unit up in the 2D plane. Vector \( \mathbf{v} = \mathbf{i} - \mathbf{j} \) translates to components \( (1, -1) \), with one unit step right and one unit down, effectively showing the influence in each direction. In a broader sense, vector components allow us to analyze and manipulate vectors by understanding their behavior in each direction independently.