Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find two vectors in opposite directions that are orthogonal to the vector \(\mathbf{u}.\) (There are many correct answers.) $$\mathbf{u}=\langle 2,6\rangle$$

Step-by-Step Solution

Answer

The two vectors that are orthogonal to the vector \(\mathbf{u}=\langle 2,6\rangle\) and in opposite directions are \(\mathbf{v}=\langle 3,-1\rangle\) and \(\mathbf{-v}=\langle -3,1\rangle\).

Step 1: Understanding Vector Dot Product

A vector \(\mathbf{v}\) is orthogonal to \( \mathbf{u} \) if and only if their dot product equals to 0. The dot product of vectors \(\mathbf{u}=\langle u_1,u_2 \rangle \) and \(\mathbf{v}=\langle v_1,v_2 \rangle \) is \( u_1*v_1 + u_2*v_2\). In this case, we have \(2*v_1 + 6*v_2 = 0\).

Step 2: Finding the Orthogonal Vectors

We need to find \(v_1\) and \(v_2\) which satisfy the above equation. A simple solution would be to set \(v_1=3\) and \(v_2=-1\). This would give us the dot product \(2*3 + 6*-1 = 0\), thus confirming that \(\mathbf{v}=\langle 3,-1\rangle \) is orthogonal to \( \mathbf{u} \).

Step 3: Finding the Opposite Vectors

The vectors in opposite directions would simply involve switching the signs of all the components. Thus the vector orthogonal and opposite to \( \mathbf{v} \) is \( \mathbf{-v} =\langle -3,1\rangle \).

Key Concepts

Vector Dot Product

The vector dot product is a fundamental concept when dealing with vectors in mathematics, especially in determining relationships such as orthogonality. In simple words, the dot product of two vectors combines them into a single number, which can provide insights into the vectors' directional relationship. If the vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \) are given, the dot product is calculated as:\\[ u_1 \cdot v_1 + u_2 \cdot v_2 \].\

• When the dot product equals zero, the vectors are orthogonal (at right angles to each other).
• In our example, find values for \( v_1 \) and \( v_2 \) such that \( 2v_1 + 6v_2 = 0 \).

Thus, finding a proper \( \mathbf{v} \) involves solving this equation, which shows the vectors' perpendicular relationship.

Opposite Vectors

Opposite vectors are intriguing as they maintain the directionality property while facing in opposing directions on the vector plane. For a vector \( \mathbf{v} = \langle v_1, v_2 \rangle \), its opposite would be \( \mathbf{-v} = \langle -v_1, -v_2 \rangle \).\

• Flipping the sign of each component results in an opposite direction.
• For example, \( \mathbf{v} = \langle 3, -1 \rangle \) becomes \( \mathbf{-v} = \langle -3, 1 \rangle \).

Thus, even if the magnitude remains the same, the orientation changes directly contrary to the original vector.

Vector Components

Vector components are the building blocks of a vector, representing projections along specified axes, typically the x and y axes in a plane. Every vector \( \mathbf{u} = \langle u_1, u_2 \rangle \) can be seen as the sum of its components along these axes.\

• The component \( u_1 \) acts along the x-axis, while \( u_2 \) aligns with the y-axis.
• To find specific behaviors such as orthogonality, we compute the relationships between these components.

In our exercise, given \( \mathbf{u} = \langle 2, 6 \rangle \), understanding and manipulating these individual components helps us discover perpendicular vectors that meet specific conditions, such as having a dot product of zero.