Question

If \((\underline{a}+\underline{b}) \cdot(\underline{a}-\underline{b})=63\) and \(|\underline{a}|=8|\underline{b}|\), then \(|\underline{a}|=\) (A) 8 (B) 64 (C) 16 4

Short answer

The value of \(|\underline{a}|\) is 8.

Step-by-step solution

Expand the given expression using the properties of dot products.

We are given that \((\underline{a}+\underline{b}) \cdot(\underline{a}-\underline{b})=63\). We will expand this expression using the properties of dot products: \((\underline{a}+\underline{b}) \cdot(\underline{a}-\underline{b}) = \underline{a}\cdot\underline{a} - \underline{a}\cdot\underline{b} + \underline{a}\cdot\underline{b} - \underline{b}\cdot\underline{b}\) We know that \(\underline{a}\cdot\underline{a}=|\underline{a}|^{2}\) and \(\underline{b}\cdot\underline{b}=|\underline{b}|^{2}\). Also, since \(\underline{a}\cdot\underline{b} = \underline{b}\cdot\underline{a}\), the middle terms will cancel out. So, we have: \(|\underline{a}|^{2} - |\underline{b}|^{2} = 63\)

Use the given relationship between \(|\underline{a}|\) and \(|\underline{b}|\) to find a relationship between their squares.

We are given \(|\underline{a}|=8|\underline{b}|\). By squaring both sides of the equation, we find: \(|\underline{a}|^{2} = 64|\underline{b}|^{2}\)

Substitute the relationship between their squares from Step 2 into the expanded expression from Step 1.

Now, substitute the relationship between \(|\underline{a}|^{2}\) and \(|\underline{b}|^{2}\) that we found in Step 2 into the expanded expression from Step 1: \(64|\underline{b}|^{2} - |\underline{b}|^{2} = 63\) Simplify the expression: \(63|\underline{b}|^{2} = 63\)

Solve for \(|\underline{b}|^{2}\) and then find \(|\underline{b}|\).

Now, solve for \(|\underline{b}|^{2}\): \(|\underline{b}|^{2} = \frac{63}{63} = 1\) Since \(|\underline{b}|\) represents the magnitude of a vector, it must be non-negative. So, we have: \(|\underline{b}| = \sqrt{1} = 1\)

Find \(|\underline{a}|\) using the relationship between \(|\underline{a}|\) and \(|\underline{b}|\).

We know that \(|\underline{a}|=8|\underline{b}|\). Substitute the value of \(|\underline{b}|\) that we found in Step 4: \(|\underline{a}| = 8(1) = 8\) So, the value of \(|\underline{a}|\) is \(\boxed{8}\) (Answer Choice \textbf{(A)}).

Key concepts

Vector Magnitude

The magnitude of a vector is essentially its length in space. It tells us how long the vector is and can be thought of as the distance from the vector's tail to its head in a geometric sense.
For a vector \( \underline{a} \), the magnitude is represented as \( |\underline{a}| \). This is calculated by taking the square root of the sum of the squares of its components.
For example, if \( \underline{a} = (x, y, z) \), then its magnitude is \( |\underline{a}| = \sqrt{x^2 + y^2 + z^2} \).
In the exercise, we are given that \( |\underline{a}| = 8|\underline{b}| \). This means that vector \( \underline{a} \) is eight times longer than vector \( \underline{b} \).
When calculating magnitudes, remember:

• The magnitude is always a non-negative value.
• It is a scalar quantity, meaning it only has magnitude, no direction.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying mathematical expressions to solve for unknown variables. It is a fundamental skill in solving equations and working with vector expressions.
In our exercise, we used algebraic manipulation to simplify the dot product equation and relate the magnitudes of vectors \( \underline{a} \) and \( \underline{b} \).

• Initially, we expanded the dot product \((\underline{a}+\underline{b}) \cdot(\underline{a}-\underline{b})\) by applying distributive properties.
• This resulted in simplification to \(|\underline{a}^2| - |\underline{b}^2| = 63\).
• We then incorporated the given relationship \(|\underline{a}| = 8|\underline{b}|\), and squared both sides to maintain equality.

These steps highlight how algebraic manipulation is pivotal to finding a solution, making it a powerful tool for breaking down complex problems into manageable steps.

Scalar Equation

A scalar equation involves expressions where the quantities can be expressed as single numeric values. They are different from vector equations that incorporate direction.
In our exercise, we transformed expressions into a scalar equation: \(64|\underline{b}|^2 - |\underline{b}|^2 = 63\).

• This was achieved by substituting the relationship \(|\underline{a}|^2 = 64|\underline{b}|^2\) into the previously simplified expression.
• Upon further simplification, it revealed \(63|\underline{b}|^2 = 63\), which is a scalar equation.
• Solving this scalar equation allowed us to find \(|\underline{b}| = 1\).

Scalar equations help simplify and solve problems with numerical accuracy. Unlike vectors, they don't include direction, focusing purely on the magnitude.