Problem 18 - Precalculus

Question

Perform the indicated vector operation, given \(u=(-4,3)\) and \(v=\langle 2,-5\rangle\) $$u-v$$

Step-by-Step Solution

Answer

The result of \(u - v\) is \((-6, 8)\).

Step 1: Understand the Problem

We need to subtract vector \(v\) from vector \(u\). This means finding the vector, \(u - v\), by subtracting the corresponding components of vectors \(u\) and \(v\).

Step 2: Identify the Components

Vector \(u\) has components \((-4, 3)\), and vector \(v\) has components \((2, -5)\). We need to use these components in our subtraction process.

Step 3: Subtract the Corresponding Components

Subtract the first component of \(v\) from the first component of \(u\), and the second component of \(v\) from the second component of \(u\): \((-4) - (2)\) and \(3 - (-5)\).

Step 4: Simplify the Components

Compute the subtraction for each component: \((-4) - 2 = -6\) \(3 - (-5) = 3 + 5 = 8\).

Step 5: Write the Resulting Vector

The resulting vector from the operation \(u - v\) is \((-6, 8)\).

Key Concepts

Vector Operations

Vector operations are mathematical procedures that involve vectors, which are entities with both magnitude and direction. They are widely used in physics and engineering to describe both real-world problems and abstract concepts. In mathematics, vectors can be manipulated similar to numbers through various operations, such as addition, subtraction, and scalar multiplication. These operations enable us to calculate and understand the relationships between vectors in geometric space. For instance, adding two vectors involves connecting them head-to-tail and finding the product of their magnitudes and directions, forming a new vector, while subtraction focuses on finding the difference in their directional components. This basic understanding is crucial when working with any vector calculations.

Component-wise Subtraction

Component-wise subtraction is a straightforward method used to subtract one vector from another. It involves calculating the difference between corresponding components of the vectors involved.

• The first step is to identify the individual components of each vector.
• Once you have the components, you subtract the first component of the second vector from the first component of the first vector.
• Then, you repeat this process for the subsequent components.

Using our example, the vector subtraction of \((-4, 3)\) and \(\langle 2, -5 \rangle\) involves calculating \((-4) - 2\) which results in \(-6\), and \(3 - (-5)\) which is equal to \(8\).
The result of this process yields the vector \((-6, 8)\). This method is essential in simplifying the process of vector subtraction, making it approachable and systematic.

Vectors

Vectors are entities represented by both a magnitude and a direction, often illustrated as arrows in a coordinate plane. They are crucial in representing quantities that cannot be described solely by a numerical value, such as velocity, force, and displacement in physics.

• Each vector consists of components, typically laid out in Cartesian coordinates, such as \((-4, 3)\) or \(\langle 2, -5 \rangle\).
• The components are usually described along the x, y, (and possibly z) axes, representing the vector's influence in each spatial direction.
• Understanding these components is vital for performing mathematical operations involving vectors.

Vectors can be expressed in different notations, such as with parentheses or angle brackets, which do not affect the operation but convey the same magnitude and direction in mathematical operations.

Resultant Vector

The resultant vector is the outcome of a vector operation, such as addition or subtraction. It combines or distinguishes the effects of two or more vectors. In our specific context, a resultant vector emerges from the component-wise subtraction of vectors.

• When subtracting vectors, the resultant vector tells us how one vector differentiates from another in terms of direction and magnitude.
• In geometric terms, this vector can be visualized as the arrow pointing from the head of the second vector to the tip of where the first vector extended.
• Another way to look at it is as a new vector that represents the net change from the initial configuration of the vectors.

For example, subtracting vector \(\langle 2, -5 \rangle\) from \((-4, 3)\) gives us the resultant vector \((-6, 8)\), portraying the cumulative change in direction from the original vectors.