Question

Show that the following "vectors " are orthogonal: \((1,-5,7,2,3) \quad\) and \((2,1,-2,7,1)\). (Hins-Consider the "dot product.")

Short answer

Vectors (1, -5, 7, 2, 3) and (2, 1, -2, 7, 1) are orthogonal because their dot product is zero.

Step-by-step solution

- Understand Orthogonality

Two vectors are orthogonal if their dot product is equal to zero. Given vectors \(\mathbf{u} = (1, -5, 7, 2, 3)\) and \(\mathbf{v} = (2, 1, -2, 7, 1)\), we need to calculate the dot product and check if it equals zero.

- Write the Dot Product Formula

The dot product of two vectors \(\mathbf{u} = (u_1, u_2, u_3, u_4, u_5)\) and \(\mathbf{v} = (v_1, v_2, v_3, v_4, v_5)\) is given by:\[ \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 + u_4 v_4 + u_5 v_5 \]

- Plug in the Values

Using the vectors given, plug in the values to calculate the dot product. \(\mathbf{u} \cdot \mathbf{v} = 1 \cdot 2 + (-5) \cdot 1 + 7 \cdot (-2) + 2 \cdot 7 + 3 \cdot 1\)

- Calculate Each Term

Calculate the individual terms in the dot product:\[ 1 \cdot 2 = 2 \]\[ (-5) \cdot 1 = -5 \]\[ 7 \cdot (-2) = -14 \]\[ 2 \cdot 7 = 14 \]\[ 3 \cdot 1 = 3 \]

- Sum the Terms

Sum all the calculated terms to get the final value of the dot product:\[ 2 + (-5) + (-14) + 14 + 3 = 0 \]

- Conclusion

Since the dot product is zero, the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

Key concepts

Dot Product

The dot product is a fundamental concept when dealing with vectors in mathematics. It allows us to find the product of two vectors and discover critical properties, such as orthogonality.

The formula for the dot product of two vectors \( \mathbf{u} = (u_1, u_2, u_3, u_4, u_5) \) and \( \mathbf{v} = (v_1, v_2, v_3, v_4, v_5) \) is given by: \[ \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 + u_4 v_4 + u_5 v_5 \]

In simple terms, you multiply the corresponding components of the two vectors and then sum these products. It's essential to practice these calculations step by step:

• Firstly, line up corresponding components from each vector.
• Next, multiply each pair of corresponding components.
• Finally, sum all these individual products.

The result of this sum tells you a lot about the relationship between the vectors, which brings us to the next concept: vector orthogonality.

Vector Orthogonality

Orthogonality is when two vectors are at right angles (90 degrees) to each other. In mathematical terms, if the dot product of two vectors is zero, then those vectors are orthogonal.

For example, consider the vectors \( \mathbf{u} = (1, -5, 7, 2, 3) \) and \( \mathbf{v} = (2, 1, -2, 7, 1) \). To check if they are orthogonal, calculate the dot product:

\[ \mathbf{u} \cdot \mathbf{v} = 1 \cdot 2 + (-5) \cdot 1 + 7 \cdot (-2) + 2 \cdot 7 + 3 \cdot 1 \]

Breaking it down:

• 1 \cdot 2 = 2
• (-5) \cdot 1 = -5
• 7 \cdot (-2) = -14
• 2 \cdot 7 = 14
• 3 \cdot 1 = 3

Now add these results:

\[ 2 + (-5) + (-14) + 14 + 3 = 0 \]

Since the sum is zero, it confirms that \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal vectors. Recognizing orthogonal vectors is important in fields like physics, computer graphics, and more.

Vector Operations

Understanding vector operations is essential to fully grasp concepts like the dot product and orthogonality. Some basic vector operations include:

• Addition: The sum of two vectors is found by adding the corresponding components. For vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), their sum is \( \mathbf{a} + \mathbf{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3) \).
• Subtraction: To subtract one vector from another, subtract each corresponding component. For example, \( \mathbf{a} - \mathbf{b} = (a_1 - b_1, a_2 - b_2, a_3 - b_3) \).
• Scalar Multiplication: Multiplying a vector by a scalar means multiplying each component by that scalar. If \( \mathbf{a} = (a_1, a_2, a_3) \) and \( k \) is a scalar, then \( k \mathbf{a} = (k \cdot a_1, k \cdot a_2, k \cdot a_3) \).

Mastering these basic operations is a stepping stone to more complex vector algebra, including understanding the dot product and orthogonal vectors. Practice these operations to gain a deeper understanding of vector mathematics and its applications.