Problem 7 - APEX Calculus

Question

In Exercises 7-16, vectors \(\vec{u}\) and \(\vec{v}\) are given. Compute \(\vec{u} \times \vec{v}\) and show this is orthogonal to both \(\vec{u}\) and \(\vec{v}\). \(\vec{u}=\langle 3,2,-2\rangle, \quad \vec{v}=\langle 0,1,5\rangle\)

Step-by-Step Solution

Answer

\(\vec{u} \times \vec{v} = \langle 12, -15, 3 \rangle\) is orthogonal to both \(\vec{u}\) and \(\vec{v}\).

Step 1: Write the Formula for the Cross Product

The cross product \( \vec{u} \times \vec{v} \) of two vectors \( \vec{u}=\langle a_1, b_1, c_1 \rangle \) and \( \vec{v}=\langle a_2, b_2, c_2 \rangle \) is given by the determinant of the matrix:\[ \vec{u} \times \vec{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \end{vmatrix} \] where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the unit vectors in the respective directions.

Step 2: Calculate the Cross Product

Substitute \( \vec{u} = \langle 3, 2, -2 \rangle \) and \( \vec{v} = \langle 0, 1, 5 \rangle \) into the cross product formula:\[ \vec{u} \times \vec{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 3 & 2 & -2 \ 0 & 1 & 5 \end{vmatrix} \]Compute the determinant:\[ \vec{u} \times \vec{v} = \mathbf{i}(2\cdot5 - (-2)\cdot1) - \mathbf{j}(3\cdot5 - (-2)\cdot0) + \mathbf{k}(3\cdot1 - 2\cdot0) \]\[ = \mathbf{i}(10 + 2) - \mathbf{j}(15) + \mathbf{k}(3) \]\[ = \mathbf{i}12 - \mathbf{j}15 + \mathbf{k}3 \]Thus, \( \vec{u} \times \vec{v} = \langle 12, -15, 3 \rangle \).

Step 3: Verify Orthogonality to \(\vec{u}\)

To show \(\vec{u} \times \vec{v}\) is orthogonal to \(\vec{u}\), compute the dot product \((\vec{u} \times \vec{v}) \cdot \vec{u}\). If the result is zero, they are orthogonal.\[ (\langle 12, -15, 3 \rangle) \cdot (\langle 3, 2, -2 \rangle) = 12\cdot3 + (-15)\cdot2 + 3\cdot(-2) \]\[ = 36 - 30 - 6 \]\[ = 0 \]Since the dot product is zero, \(\vec{u} \times \vec{v}\) is orthogonal to \(\vec{u}\).

Step 4: Verify Orthogonality to \(\vec{v}\)

To show \(\vec{u} \times \vec{v}\) is orthogonal to \(\vec{v}\), compute the dot product \((\vec{u} \times \vec{v}) \cdot \vec{v}\). If the result is zero, they are orthogonal.\[ (\langle 12, -15, 3 \rangle) \cdot (\langle 0, 1, 5 \rangle) = 12\cdot0 + (-15)\cdot1 + 3\cdot5 \]\[ = 0 - 15 + 15 \]\[ = 0 \]Since the dot product is zero, \(\vec{u} \times \vec{v}\) is orthogonal to \(\vec{v}\).

Key Concepts

Orthogonal Vectors

Two vectors are said to be orthogonal if they are perpendicular to each other.
This means their dot product equals zero. If you imagine these vectors as arrows, they would form a 90-degree angle at their intersection.
When working with vectors in three dimensions, this concept of orthogonality becomes useful in many areas such as physics, computer graphics, and engineering.

• When computing the cross product of two vectors, the resulting vector is always orthogonal to both original vectors.
• Let's consider vectors \( \vec{u} \) and \( \vec{v} \), the cross product \( \vec{u} \times \vec{v} \) results in a vector that is orthogonal to both \( \vec{u} \) and \( \vec{v} \).

Orthogonality is a key property that is exploited in various mathematical applications and solutions. By confirming zero dot product, we verify that vectors are truly orthogonal in the space.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number.
It is a measure of how much one vector extends in the direction of another.
The formula for the dot product of two 3D vectors \( \vec{a} = \langle a_1, b_1, c_1 \rangle \) and \( \vec{b} = \langle a_2, b_2, c_2 \rangle \) is:\[ \vec{a} \cdot \vec{b} = a_1a_2 + b_1b_2 + c_1c_2 \]

• If the dot product is zero, the vectors are orthogonal.
• The dot product is useful in determining angles between vectors.

In the context of verifying orthogonality after a cross product, the dot product confirms the perpendicularity of vectors. Without the dot product, checking if vectors align 90-degree to each other would be more challenging.

Vector Mathematics

Vector mathematics is a powerful tool in both geometry and physics, enabling complex calculations with ease. Vectors are quantities that have both magnitude and direction.
They can represent anything from forces to velocities, making them incredibly versatile.
In mathematical terms, vectors are often denoted with symbols like \( \vec{u} \) and \( \vec{v} \) and can be represented in component form such as \( \langle a, b, c \rangle \).

• Cross product is one of the fundamental operations in vector mathematics, resulting in a vector that is orthogonal to the original two.
• This operation is only applicable in three-dimensional space, unlike the dot product which works in any dimension.

Using cross product and the corresponding verification using dot product, we can solve real-world problems requiring vector orthogonal computation.