Problem 61 - Precalculus with Limits

Question

Vector Operations In Exercises \(57-62\) , find the component form of \(v\) and sketch the specified vector operations geometrically, where \(u=2 i-j\) and \(w=i+2 j\) $$\mathbf{v}=\frac{1}{2}(3 \mathbf{u}+\mathbf{w})$$

Step-by-Step Solution

Answer

The component form of the vector v is \(v = 3.5i - 0.5j \). This vector starts at origin (0, 0) and ends at the point (3.5, -0.5) in the 2D-space.

Step 1: Identify the values given in the problem

The initial vectors are given as u=2i - j and w=i + 2j. The vector v is defined as v = 0.5*(3*u + w).

Step 2: Calculate the arithmetic of vectors

The calculation of the vector v can be carried out as follows: \[v = 0.5*(3*(2i - j) + (i + 2j))\], which simplifies to: \[v = 0.5*(6i - 3j + i + 2j)\], and finally after collecting like terms: \[v = 0.5 * (7i - j)\].

Step 3: Simplify the vector

Finally, we simplify v by carrying out the multiplication across the vector: \[v = 3.5i - 0.5j\] .

Step 4: Plotting the vector

Vector v as calculated can be represented geometrically as an arrow starting from origin (0, 0) ending at the point (3.5, -0.5). The i-component forms the x-coordinate and the j-component forms the y-coordinate of the end point of the vector arrow. The arrow is drawn from the origin to this point.

Key Concepts

Component Form

Understanding the component form of vectors is crucial in manipulating and analyzing vectors quantitatively. Vectors like \( \mathbf{u} = 2i - j \) and \( \mathbf{w} = i + 2j \) are expressed in their component forms. The component form of a vector describes it using its horizontal and vertical components, in this case, represented by the coefficients of \( i \) and \( j \).
Each vector can be thought of as a directed line from the origin of a coordinate system. Its component in the direction of \( i \) (horizontal axis) is often seen as the vector's x-component, while its component in the direction of \( j \) (vertical axis) is the vector's y-component.

• For the vector \( \mathbf{u} = 2i - j \), the x-component is 2, and the y-component is -1.
• For the vector \( \mathbf{w} = i + 2j \), the x-component is 1, and the y-component is 2.
• The resulting vector \( \mathbf{v} = 3.5i - 0.5j \) has an x-component of 3.5 and a y-component of -0.5.

Hence, when we write out vectors in their component forms, it becomes extremely intuitive to conduct further vector operations. This yields a clear pathway for calculating and simplifying vector results.

Geometric Representation

Geometrically representing vectors helps in visualizing them, which is vital for deeper understanding. Vectors can be represented on a Cartesian plane as arrows from the origin to a determined point. This point is given by the vector's component form.
For example, vector \( \mathbf{v} = 3.5i - 0.5j \) is represented by the point (3.5, -0.5) on the coordinate grid. The tail of the vector arrow begins at the origin \( (0,0) \), and the head lands at point \( (3.5, -0.5) \).
Here are some important geometric aspects of vector representation:

• The length of the arrow, also known as the magnitude, is determined by both the components. Use the Pythagorean theorem: \( \sqrt{(x^2 + y^2)} \) to calculate it, where \( x \) and \( y \) are the components.
• The direction is indicated by the way the arrow is pointing, and it can also be expressed as an angle from the horizontal axis.
• By drawing multiple vectors, you can also visually represent vector addition, subtraction, and more complex operations.

Understanding these basics can greatly enhance your ability to work with vectors in both abstract and real-world applications.

Arithmetic of Vectors

Vector arithmetic involves operations such as addition, subtraction, and scalar multiplication. In this exercise, various arithmetic steps were taken to determine vector \( \mathbf{v} \).

• Addition: When adding vectors, you simply add their respective components. For instance, adding \( 3\mathbf{u} = 6i - 3j \) and \( \mathbf{w} = i + 2j \), results in \( (6i + i) + (-3j + 2j) = 7i - j \).
• Scalar Multiplication: When a vector is multiplied by a scalar, each component is multiplied by the scalar. Thus, the vector \( 0.5 \times (7i - j) \) results in \( (0.5 \times 7)i + (0.5 \times -1)j = 3.5i - 0.5j \).

These operations adhere to regular algebraic laws, allowing you to work out even complicated expressions step-by-step in a manageable way. The ability to navigate through vector arithmetic opens up the potential to model numerous physical phenomena, where vectors naturally occur.