Problem 16 - Algebra and Trigonometry

Question

Find the vectors \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},\) and \(3 \mathbf{u}-\frac{1}{2} \mathbf{v}\) $$ \mathbf{u}=\langle 0,1,-3\rangle, \mathbf{v}=\langle 4,2,0\rangle $$

Step-by-Step Solution

Answer

\( \mathbf{u}+\mathbf{v} = \langle 4, 3, -3 \rangle, \mathbf{u}-\mathbf{v} = \langle -4, -1, -3 \rangle, 3\mathbf{u}-\frac{1}{2}\mathbf{v} = \langle -2, 2, -9 \rangle \).

Step 1: Calculate \( \mathbf{u} + \mathbf{v} \)

To find \( \mathbf{u} + \mathbf{v} \), add the corresponding components of \( \mathbf{u} \) and \( \mathbf{v} \): \( \langle 0+4, 1+2, -3+0 \rangle = \langle 4, 3, -3 \rangle \).

Step 2: Calculate \( \mathbf{u} - \mathbf{v} \)

To find \( \mathbf{u} - \mathbf{v} \), subtract the corresponding components of \( \mathbf{v} \) from \( \mathbf{u} \): \( \langle 0-4, 1-2, -3-0 \rangle = \langle -4, -1, -3 \rangle \).

Step 3: Calculate \( 3\mathbf{u} - \frac{1}{2} \mathbf{v} \)

First, scale \( \mathbf{u} \) by 3: \( 3\mathbf{u} = \langle 3 \times 0, 3 \times 1, 3 \times (-3) \rangle = \langle 0, 3, -9 \rangle \). Then, scale \( \mathbf{v} \) by \( \frac{1}{2} \): \( \frac{1}{2}\mathbf{v} = \langle 2, 1, 0 \rangle \). Compute \( 3\mathbf{u} - \frac{1}{2}\mathbf{v} = \langle 0-2, 3-1, -9-0 \rangle = \langle -2, 2, -9 \rangle \).

Key Concepts

Understanding Vector Operations

Vector operations such as addition and subtraction play a crucial role in mathematics and physics. They help us understand the manipulation of quantities that have both magnitude and direction. In the realm of vector operations, we focus primarily on adding and subtracting vectors. These operations involve specific rules regarding the combination of their components.

**Vector Addition**: When adding two vectors, we align their respective components. For instance, given \( \mathbf{u} = \langle 0, 1, -3 \rangle \) and \( \mathbf{v} = \langle 4, 2, 0 \rangle \), we add them by summing the corresponding elements:

• Add the x-components: \( 0 + 4 = 4 \)
• Add the y-components: \( 1 + 2 = 3 \)
• Add the z-components: \( -3 + 0 = -3 \)

The resulting vector is \( \mathbf{u} + \mathbf{v} = \langle 4, 3, -3 \rangle \).

**Vector Subtraction**: To subtract vectors, you follow a similar process as addition but instead, subtract each of the components. For instance:

• Subtract x-components: \( 0 - 4 = -4 \)
• Subtract y-components: \( 1 - 2 = -1 \)
• Subtract z-components: \( -3 - 0 = -3 \)

The result of subtraction gives us \( \mathbf{u} - \mathbf{v} = \langle -4, -1, -3 \rangle \).

Exploring Vector Components

Each vector is composed of components that determine its direction and magnitude in space. Understanding these components is essential when performing operations like addition, subtraction, or scaling.

**Components of a Vector**: In an n-dimensional space, a vector can be represented as \( \langle x, y, z, \ldots \rangle \) with each value being a component of the vector. These components represent the vector's projection along each axis in the specified coordinate system. For example, the vector \( \mathbf{u} = \langle 0, 1, -3 \rangle \) consists of:

• x-component: 0
• y-component: 1
• z-component: -3

The components are fundamental because they allow us to perform vector operations by dealing with each corresponding part separately. This component-wise manipulation ensures that vector mathematics adheres to geometric intuition and physical principles.

Understanding these components helps us visualize vectors as arrows in space, making it easier to grasp how vectors behave when transformed through operations.

The Concept of Scalar Multiplication

Scalar multiplication refers to the process of enlarging or shrinking a vector by a scalar value. When a vector is multiplied by a scalar, each component of the vector is scaled by that scalar.

**How Scalar Multiplication Works**: Given a vector \( \mathbf{v} = \langle x, y, z \rangle \) and a scalar \( k \), the result of scalar multiplication is \( k\mathbf{v} = \langle kx, ky, kz \rangle \). This allows for flexible adjustments to the vector's magnitude without altering its direction unless the scalar is negative, in which case, the direction is reversed.

**Example**: Consider \( 3\mathbf{u} \) where \( \mathbf{u} = \langle 0, 1, -3 \rangle \). Scaling \( \mathbf{u} \) by 3 yields:

• x-component: \( 3 \times 0 = 0 \)
• y-component: \( 3 \times 1 = 3 \)
• z-component: \( 3 \times (-3) = -9 \)

Scalar multiplication is beneficial as it accommodates vector modifications in various situations, ranging from physical simulations to graphical transformations. Moreover, when combined with other operations, scalar multiplication can solve complex problems elegantly, as seen with the expression \( 3\mathbf{u} - \frac{1}{2}\mathbf{v} \). Here, each vector is first scaled, then the results are computed through addition and subtraction.