Problem 66 - 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.) $$\mathbf{u}=\langle-7,5\rangle$$

Step-by-Step Solution

Answer

The two orthogonal vectors that are in opposite directions to \(\mathbf{u}=\langle-7,5\rangle\) are \(\mathbf{v}=\langle5,7\rangle\) and \(\mathbf{w}=\langle-5,-7\rangle\).

Step 1: Understand the Properties of Orthogonal Vectors

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal if and only if their dot product equals zero. That is, \(\mathbf{u}\cdot\mathbf{v}=0\). A useful point to note is that if \(\mathbf{u}=\langle a, b \rangle\), then any orthogonal vector will be of the form \(\langle b, -a \rangle\) or \(\langle -b, a\rangle\).

Step 2: Find the Orthogonal Vector

Given the vector \(\mathbf{u}=\langle-7,5\rangle\), an orthogonal vector can be found by reversing the order of the elements and changing one of their signs. By this rule, one of the orthogonal vectors can be \(\mathbf{v}=\langle 5,7\rangle\).This vector \(\mathbf{v}\) is therefore orthogonal to \(\mathbf{u}\). Using the dot product, \(\mathbf{u}\cdot\mathbf{v}=(-7 \cdot 5) +(5 \cdot 7)=0\), thus confirming the orthogonality of \(\mathbf{u}\) and \(\mathbf{v}\).

Step 3: Find the Opposite Vector

To find a vector in the opposite direction of \(\mathbf{v}=\langle5,7\rangle\), just switch the signs of its components. Hence, the opposite vector of \(\mathbf{v}\) is \(\mathbf{w}=\langle-5,-7\rangle\). This vector \(\mathbf{w}\) is also orthogonal to \(\mathbf{u}\), as their dot product equals zero: \(\mathbf{u}\cdot\mathbf{w}=(-7 \cdot -5) +(5 \cdot -7)=0\).

Key Concepts

Dot Product

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Vector Orthogonality

Vector orthogonality is a concept describing the perpendicularity of vectors. Two vectors are orthogonal if the angle between them is 90 degrees. The significance of this lies in their dot product: if two vectors are orthogonal, their dot product equals exactly zero.

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Opposite Direction Vectors

Vectors in opposite directions can be thought of as arrows pointing in completely reversed paths. Conceptually, if one vector points to the right, the other would point just as far to the left. To find such opposite vectors mathematically, one simply negates each component of the original vector, resulting in a vector with the same magnitude but opposite direction.

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