Problem 21 - University Calculus: Early Transcendentals

Question

Express each vector in the form \(\mathbf{w}=w_{1} \mathbf{i}+\) \(w_{2} \mathbf{j}+w_{3} \mathbf{k}\). \(5 \mathbf{u}-\mathbf{v}\) if \(\mathbf{u}=\langle 1,1,-1\rangle\) and \(\mathbf{v}=\langle 2,0,3\rangle\)

Step-by-Step Solution

Answer

\( \mathbf{w} = 3\mathbf{i} + 5\mathbf{j} - 8\mathbf{k} \)

Step 1: Multiply Vector u by 5

First, multiply each component of vector \( \mathbf{u} = \langle 1, 1, -1 \rangle \) by 5. This gives us: \( 5 \cdot \mathbf{u} = 5 \langle 1, 1, -1 \rangle = \langle 5, 5, -5 \rangle \).

Step 2: Write Down Vector v

Now, let’s write down the components of vector \( \mathbf{v} \), which is already given as \( \langle 2, 0, 3 \rangle \).

Step 3: Subtract Vector v from 5u

Now, subtract the components of vector \( \mathbf{v} \) from the components of \( 5\mathbf{u} \). This gives: \[ \mathbf{w} = \langle 5 - 2, 5 - 0, -5 - 3 \rangle = \langle 3, 5, -8 \rangle. \]

Step 4: Express the Resulting Vector

Express the vector \( \mathbf{w} = \langle 3, 5, -8 \rangle \) in the form of \( w_{1} \mathbf{i} + w_{2} \mathbf{j} + w_{3} \mathbf{k} \). So, \( \mathbf{w} = 3\mathbf{i} + 5\mathbf{j} - 8\mathbf{k} \).

Key Concepts

Vector Operations

In vector calculus, vector operations are fundamental processes that involve manipulating vectors. Vectors are mathematical entities characterized by both magnitude and direction. They can be represented in various forms including component form, unit vector notation, or magnitude-angle form. One of the primary operations in vectors is addition and subtraction, often crucial in physics and engineering when dealing with forces or velocities. When performing such operations, it is important to understand how vectors behave under different scenarios.

• Addition involves aligning vectors head-to-tail and then joining the initial point of the first vector directly to the terminal point of the last.
• Subtraction, in turn, involves adding the inverse of a vector to another, effectively "reversing" the direction of the vector to be subtracted.
• Additionally, scalar multiplication, another vector operation, involves scaling a vector by a real number, changing its magnitude but not its direction, unless the scalar is negative.

Mastering vector operations paves the way for tackling more advanced problems involving any number of multi-dimensional vectors.

Vector Subtraction

Vector subtraction may seem complex, but it's merely an extension of basic arithmetic. Simply put, when you subtract one vector from another, you are effectively creating a new vector that bridges the gap between their endpoint differences. The process requires subtracting the corresponding components of two given vectors in sequence. For instance, if you have vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \),
the difference is computed as:
\[ \mathbf{a} - \mathbf{b} = \langle a_1 - b_1, a_2 - b_2, a_3 - b_3 \rangle \]
Subtraction in vector terms usually represents a directional change or indicates a shift in position.
Always visualize subtraction as repositioning the initial vector in a system, allowing the design of resultant vectors needed for solutions in vector problems.

Scalar Multiplication

Scalar multiplication is a crucial operation when it comes to modifying the size of vectors. By multiplying a vector by a scalar (a real number), you alter its magnitude without changing its direction. This operation resizes the vector's components proportionally to the scalar factor.
For example, if we have vector \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and a scalar \( c \), then multiplying them will yield:
\[ c \cdot \mathbf{u} = \langle cu_1, cu_2, cu_3 \rangle \]
In scenarios where the scalar is negative, the direction of the vector reverses, although its length changes accordingly. Scalar multiplication plays a pivotal role in defining the magnitude of vectors in various physics and engineering problems, such as scaling force vectors or determining resultant directions in motion plans.

Vector Components

Understandably, grasping vector components is key to solving many problems involving vectors. Vector components refer to the projections of a vector along the axes of a coordinate system. For a three-dimensional vector \( \mathbf{w} = \langle w_1, w_2, w_3 \rangle \), it's expressed in unit vector form as \( w_1 \mathbf{i} + w_2 \mathbf{j} + w_3 \mathbf{k} \).

The components highlight how much of the vector falls along the x, y, and z-axis respectively. This representation not only simplifies calculations involving vectors but also helps in visualizing spatial relationships.

• To break down vectors into components, understand that each component is calculated by the product of the vector's magnitude and the cosine of the angle it forms with the respective axis.
• They are essential for performing addition, subtraction, and scalar products in multi-dimensional spaces.

Working with vector components streamlines solving vector problems, especially in physics, where forces need decomposing into effective directional parts.