Problem 71 - Precalculus

Question

Given \(u=\langle a, b\rangle\) and \(v=\langle c, d\rangle,\) show that the following properties are true: $$0 \cdot u=0$$

Step-by-Step Solution

Answer

Multiply each component of the vector by 0 to get the zero vector \( \langle 0, 0 \rangle \).

Step 1: Understand the Terms

We are given two vectors, \( u = \langle a, b \rangle \) and \( v = \langle c, d \rangle \). Our task is to demonstrate that the scalar multiplication of zero by the vector \( u \) results in the zero vector \( \langle 0, 0 \rangle \).

Step 2: Apply Scalar Multiplication Definition

The scalar multiplication of a vector \( u = \langle a, b \rangle \) by a scalar \( k \) is obtained by multiplying each component of the vector by \( k \). Therefore, \( k \cdot u = k \cdot \langle a, b \rangle = \langle k \cdot a, k \cdot b \rangle \).

Step 3: Substitute the Scalar with Zero

Now, substitute \( k = 0 \) into the equation from Step 2. Thus, we have \( 0 \cdot u = \langle 0 \cdot a, 0 \cdot b \rangle \).

Step 4: Simplify the Expression

We know that \( 0 \cdot a = 0 \) and \( 0 \cdot b = 0 \), therefore, \( \langle 0 \cdot a, 0 \cdot b \rangle = \langle 0, 0 \rangle \). This is the zero vector.

Step 5: Conclusion

Since \( 0 \cdot u = \langle 0, 0 \rangle \), we have proven that multiplying a vector by zero results in the zero vector. Thus, the property \( 0 \cdot u = 0 \) is true.

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in vector algebra. It involves multiplying each component of a vector by the same scalar value, which is a number. Here, the vector, let's say \( u = \langle a, b \rangle \), is multiplied by a scalar \( k \). The result of this multiplication is another vector, \( k \cdot u = \langle k \cdot a, k \cdot b \rangle \).

By doing this:

• Each component of the vector \( u \) is scaled by the amount \( k \).
• The direction of the vector stays the same if \( k \) is positive, but reverses if \( k \) is negative.
• The length of the vector is scaled by the absolute value of \( k \).

This process is linear, meaning it respects the principles of vector addition and scalar multiplication terms. Understanding scalar multiplication forms the basis for more complex vector operations.

Zero Vector

The zero vector plays a crucial role in vector mathematics. It is represented as \( \langle 0, 0 \rangle \) in two-dimensional space but extends similarly into higher dimensions. This vector is unique as it holds special properties:

- It has no direction because its magnitude is zero.
- It acts as the additive identity in vector addition, meaning any vector added to it remains unchanged.
- When a vector is multiplied by zero, the result is the zero vector, consistent with our previous multiplication example.

Understanding the zero vector is essential as it frequently appears in solving vector equations and understanding vector spaces.

Vector Properties

Vectors have several important properties that define how they interact under various operations. These properties are essential for anyone working with vector math.

1. **Commutative Property of Addition:** A vector addition is commutative. This means \( u + v = v + u \). The order of addition doesn’t matter here.
2. **Associative Property of Addition:** Vectors can be grouped in any way in addition without changing the result. Thus, \( (u + v) + w = u + (v + w) \).
3. **Distributive Property:** Scalar multiplication distributes over vector addition as well as over scalar addition, i.e., \( k(u + v) = ku + kv \) and \((k + m)u = ku + mu \).
4. **Zero Vector Identity:** Any vector \( u \) plus the zero vector results in \( u \) itself, expressed as \( u + 0 = u \).
5. **Scalar Multiplication Identity:** Multiplying any vector by 1 gives the vector itself, i.e., \( 1 \cdot u = u \).

These properties form the foundation of vector calculus and algebra, enabling complex computations and insights into physical problems.