Problem 47 - Algebra and Trigonometry

Question

Given vectors \(\mathbf{u}=2 x \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{v}=x \mathbf{i}-8 \mathbf{j},\) find \(x\) so that \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

Step-by-Step Solution

Answer

The value of \( x \) is \( \pm 2 \sqrt{3} \).

Step 1: Understand Orthogonality Condition

Two vectors \(\textbf{u}\) and \(\textbf{v}\) are orthogonal if their dot product is zero. This means \(\textbf{u} \bullet \textbf{v} = 0\).

Step 2: Write the Dot Product Formula

The dot product of two vectors \(\textbf{u}\) and \(\textbf{v}\) is given by \[\mathbf{u} \bullet \textbf{v} = (2x) \bullet x + (3) \bullet (-8).\]

Step 3: Simplify the Dot Product

Substitute the components of \(\textbf{u}\) and \(\textbf{v}\) into the above formula to get: \[2x^2 + 3(-8) = 0\]

Step 4: Solve the Equation

Solve the equation \[2x^2 + 3(-8) = 0\] to get \[2x^2 - 24 = 0\]. Add 24 to both sides: \[2x^2 = 24\]. Divide both sides by 2: \[x^2 = 12\]. Finally, take the square root of both sides: \[x = \pm \sqrt{12} = \pm 2 \sqrt{3}\].

Key Concepts

Dot Product

The dot product is a fundamental concept in vector algebra. It's a way to multiply two vectors and obtain a scalar (a single number). This operation combines elements of the vectors following specific rules. If you have two vectors, \(\textbf{a} \) and \(\textbf{b} \), the dot product, written as \(\textbf{a} \bullet \textbf{b} \), is calculated as:
\[ \mathbf{a} \bullet \mathbf{b} = a_1b_1 + a_2b_2 + ... + a_nb_n \] Here, \(\textbf{a} \) and \(\textbf{b} \) are vectors with components \(a_1, a_2, ..., a_n \) and \(\textbf{b}_1, \textbf{b}_2, ..., \textbf{b}_n \), respectively. The dot product is commutative, meaning \(\textbf{a} \bullet \textbf{b} = \textbf{b} \bullet \textbf{a} \).
In our exercise, we have vectors \(\textbf{u} = 2x \textbf{i} + 3 \textbf{j} \) and \(\textbf{v} = x \textbf{i} - 8 \textbf{j} \). To find their dot product, we multiply corresponding components and sum them up:
\[ \textbf{u} \bullet \textbf{v} = (2x \bullet x) + (3 \bullet -8) \] This simplifies to:
\[ \textbf{u} \bullet \textbf{v} = 2x^2 - 24 \]

Vector Computation

Vectors are essential in geometry, physics, and many fields of engineering and computer science. A vector is defined by its magnitude (length) and direction. In two dimensions, vectors are often represented as \(\textbf{v} = a \textbf{\textbf{i}} + b \textbf{\textbf{j}} \), where \(a \) and \(b \) are components along the x and y axes, respectively.
In the context of our exercise, we start with vectors \(\textbf{u} = 2x \textbf{i} + 3 \textbf{j} \) and \(\textbf{v} = x \textbf{i} - 8 \textbf{j} \). Each component, \(\textbf{i} \) and \(\textbf{j} \), represents a unit vector along the x-axis and y-axis. To compute the dot product, alignment of these components is crucial.
Follow these steps for vector computation:

• Identify each vector's components.
• Multiply corresponding components (x with x and y with y).
• Sum these products to get the dot product.

Here, we multiply and sum the components of \(\textbf{u} \) and \(\textbf{v} \) to get:
\[ (2x)x + (3)(-8) = 2x^2 - 24 \]

Orthogonality Condition

Two vectors are orthogonal if they meet at a right angle (90 degrees). This means their dot product is zero. The orthogonality condition is therefore written as:
\[ \textbf{u} \bullet \textbf{v} = 0 \] This is significant in various applications such as computer graphics, where determining perpendicular vectors is crucial.
In our exercise, vectors \(\textbf{u} \) and \(\textbf{v} \) are orthogonal if:
\[ 2x^2 - 24 = 0 \] To find \(x \), solve the equation:

• Add 24 to both sides: \ 2x^2 = 24 \
• Divide by 2: \ x^2 = 12 \
• Take the square root: \ x = \pm 2 \sqrt{3} \

Thus, \( x = \pm 2 \sqrt{3} \) satisfies the orthogonality condition, making vectors \(\textbf{u} \) and \(\textbf{v} \) perpendicular.