Question

Show that two nonzero vectors \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle\) and \(\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\) are perpendicular to each other if \(u_{1} v_{1}+u_{2} v_{2}=0\)

Short answer

Answer: The two vectors are perpendicular to each other.

Step-by-step solution

Recall the dot product formula for 2D vectors

The dot product of two vectors \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle\) and \(\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\) is given by: \(\mathbf{u} \cdot \mathbf{v} = u_{1}v_{1} + u_{2}v_{2}\)

Check the given condition

We are given that \(u_{1} v_{1}+u_{2} v_{2}=0\), which is the same as the dot product \(\mathbf{u} \cdot \mathbf{v}\).

Relate the dot product to perpendicularity

Two nonzero vectors are perpendicular to each other when their dot product is zero. So, if \(\mathbf{u} \cdot \mathbf{v} = 0\), then the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular.

Conclusion

Since our given condition is equivalent to the dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) being zero, we can conclude that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular to each other when \(u_{1} v_{1}+u_{2} v_{2}=0\).

Key concepts

Perpendicular Vectors

Perpendicular vectors in mathematics are vectors that intersect at a 90-degree angle. This special relationship can help in many applications, including physics and engineering. When two vectors are perpendicular, it means they do not influence each other in the direction of the other vector.
This concept is mathematically defined by the dot product. Two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \) are perpendicular if their dot product is zero.

• They are at a right angle to each other.
• They do not share directional influence.

Recognizing perpendicular vectors is crucial for simplifying vector calculations. It helps in determining orthogonality and assessing the independence of vectors. For example, in coordinate geometry, the x-axis and y-axis are perpendicular, providing two independent directions to describe any point on the plane.

2D Vectors

Two-dimensional (2D) vectors are essential in representing quantities that have both magnitude and direction within a plane. They are often depicted as arrows in the Cartesian coordinate system with components \( \langle u_1, u_2 \rangle \). These components describe exactly how far and in what direction the vector moves.
Understanding 2D vectors is vital for interpreting motion, forces, and various physical phenomena described in two dimensions.

• Direction: Represented by the angle of the vector relative to a reference direction.
• Magnitude: Length of the vector, calculated using the Pythagorean theorem \( \sqrt{u_1^2 + u_2^2} \).

2D vectors are often easier to handle because they only require two components. They are frequently used in physics to study objects in a plane, helping to analyze forces or velocities.

Dot Product Formula

The dot product, also known as the scalar product, is a fundamental operation used to multiply two vectors. For 2D vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), the dot product is \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \).

• The result of a dot product is a scalar, not a vector.
• Useful for determining vector projection and the angle between vectors.

One key property of the dot product is detecting perpendicularity. Two vectors are perpendicular if their dot product is zero. This is a quick method to test for orthogonality between vectors. The dot product also relates to the cosine of the angle \( \theta \) between two vectors: \( \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta \). When vectors are perpendicular, \( \cos \theta = 0 \), reaffirming that the dot product is zero.