Problem 13 - APEX Calculus

Question

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Compute \(\vec{u} \times \vec{v}\) and show this is orthogonal to both \(\vec{u}\) and \(\vec{v}\). \(\vec{u}=\langle a, b, 0\rangle, \quad \vec{v}=\langle c, d, 0\rangle\)

Step-by-Step Solution

Answer

The cross product is \( \langle 0, 0, ad-bc \rangle \), and it is orthogonal to both \( \vec{u} \) and \( \vec{v} \).

Step 1: Compute the Cross Product

To find the cross product \( \vec{u} \times \vec{v} \) for the vectors \( \vec{u} = \langle a, b, 0 \rangle \) and \( \vec{v} = \langle c, d, 0 \rangle \), we use the formula for the cross product of two 3D vectors: \[ \vec{u} \times \vec{v} = \langle u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1 \rangle \]Substitute \( u_3 = 0 \) and \( v_3 = 0 \) into the formula:\[ \vec{u} \times \vec{v} = \langle b \cdot 0 - 0 \cdot d, 0 \cdot c - a \cdot 0, a \cdot d - b \cdot c \rangle \]This simplifies to:\[ \vec{u} \times \vec{v} = \langle 0, 0, ad - bc \rangle \].

Step 2: Check Orthogonality with \( \vec{u} \)

A vector is orthogonal to another if their dot product is zero. We need to check if \( \vec{u} \times \vec{v} = \langle 0, 0, ad-bc \rangle \) is orthogonal to \( \vec{u} = \langle a, b, 0 \rangle \):Compute the dot product:\[ \langle a, b, 0 \rangle \cdot \langle 0, 0, ad-bc \rangle = a \cdot 0 + b \cdot 0 + 0 \cdot (ad-bc) = 0 \].Since the dot product is zero, \( \vec{u} \times \vec{v} \) is orthogonal to \( \vec{u} \).

Step 3: Check Orthogonality with \( \vec{v} \)

Similarly, we check if \( \vec{u} \times \vec{v} = \langle 0, 0, ad-bc \rangle \) is orthogonal to \( \vec{v} = \langle c, d, 0 \rangle \):Compute the dot product:\[ \langle c, d, 0 \rangle \cdot \langle 0, 0, ad-bc \rangle = c \cdot 0 + d \cdot 0 + 0 \cdot (ad-bc) = 0 \].Since the dot product is zero, \( \vec{u} \times \vec{v} \) is orthogonal to \( \vec{v} \).

Key Concepts

Orthogonality

Orthogonality is an interesting concept in the world of vectors. Two vectors are said to be orthogonal if they are perpendicular to each other, meaning they meet at a right angle (90 degrees). This property is particularly useful in mathematics and physics, where it helps in simplifying calculations and understanding vector spaces.

In this context, orthogonality is determined using the dot product. When the dot product of two vectors is zero, it indicates that the vectors are orthogonal. For the vectors \( \vec{u} = \langle a, b, 0 \rangle \) and \( \vec{v} = \langle c, d, 0 \rangle \), the cross product \( \vec{u} \times \vec{v} = \langle 0, 0, ad-bc \rangle \) is orthogonal to both \( \vec{u} \) and \( \vec{v} \). This is confirmed by the dot product calculations, which yield zero for both vectors.

This orthogonality implies that the resulting cross product vector serves as a normal or perpendicular to the plane formed by \( \vec{u} \) and \( \vec{v} \). Understanding this concept helps students recognize how properties of vectors like direction and magnitude affect spatial relationships.

Vector Operations

Vector operations are fundamental to working with vectors in mathematics and physics. They involve processes such as addition, subtraction, and multiplication. These operations allow you to find new vectors that have different properties and applications.

A key operation is the cross product, which produces a new vector that is perpendicular to the original two vectors. This operation is denoted by \( \vec{u} \times \vec{v} \) and is defined for three-dimensional vectors. In this exercise, we calculate the cross product of \( \vec{u} = \langle a, b, 0 \rangle \) and \( \vec{v} = \langle c, d, 0 \rangle \), resulting in \( \langle 0, 0, ad-bc \rangle \).

Another crucial vector operation is the dot product, which results in a scalar and indicates whether two vectors are orthogonal. Through engaging with vector operations, students gain skills in manipulating vectors to uncover meaningful relationships in three-dimensional space.

Dot Product

The dot product, also known as the scalar product, is a simple yet powerful tool in vector mathematics. It takes two vectors and returns a single number or scalar.

The formula for the dot product of two vectors \( \vec{u} = \langle u_1, u_2, u_3 \rangle \) and \( \vec{v} = \langle v_1, v_2, v_3 \rangle \) is:\[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \]

In our example, the cross product \( \vec{u} \times \vec{v} = \langle 0, 0, ad-bc \rangle \) is orthogonal to \( \vec{u} \). We verify this by computing the dot product, which results in zero:\[ \langle a, b, 0 \rangle \cdot \langle 0, 0, ad-bc \rangle = 0 \]

Similarly, for \( \vec{v} \):\[ \langle c, d, 0 \rangle \cdot \langle 0, 0, ad-bc \rangle = 0 \]

A dot product of zero proves orthogonality between vectors. By understanding how to calculate and interpret the dot product, students develop a better grasp of aligning and positioning vectors in space.