Question

If two vectors \(\vec{a}\) and \(\vec{b}\) are such that \(|\vec{a}| = 2\), \(|\vec{b}| = 3\) and \(\vec{a} \cdot \vec{b} = 4\), find \(|\vec{a} - \vec{b}|\).

Hint:

This is a very common type of vector problem. Remember the two key expansion formulas:
\(|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + 2(\vec{a} \cdot \vec{b}) + |\vec{b}|^2\)
\(|\vec{a} - \vec{b}|^2 = |\vec{a}|^2 - 2(\vec{a} \cdot \vec{b}) + |\vec{b}|^2\)
These are analogous to the algebraic identities \((x+y)^2\) and \((x-y)^2\).

Answer 1

Step 1: Understanding the Concept:
The problem asks for the magnitude of the difference between two vectors, given their individual magnitudes and their dot product. The key is to relate the magnitude of a vector to the dot product of the vector with itself.
Step 2: Key Formula or Approach:
We will use the property that for any vector \(\vec{v}\), its magnitude squared is given by \(|\vec{v}|^2 = \vec{v} \cdot \vec{v}\).
1. Apply this property to the vector \((\vec{a} - \vec{b})\): \(|\vec{a} - \vec{b}|^2 = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b})\).
2. Expand the dot product using its distributive property.
3. Substitute the given values of \(|\vec{a}|\), \(|\vec{b}|\), and \(\vec{a} \cdot \vec{b}\).
4. Take the square root to find \(|\vec{a} - \vec{b}|\).
Step 3: Detailed Explanation or Calculation:
We want to find \(|\vec{a} - \vec{b}|\). Let's start by calculating its square: \[ |\vec{a} - \vec{b}|^2 = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) \] Expand the dot product: \[ = \vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{b} - \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b} \] Using the properties \(\vec{v} \cdot \vec{v} = |\vec{v}|^2\) and the commutative property of the dot product (\(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\)): \[ = |\vec{a}|^2 - 2(\vec{a} \cdot \vec{b}) + |\vec{b}|^2 \] Now, substitute the given values: \(|\vec{a}| = 2\), \(|\vec{b}| = 3\), and \(\vec{a} \cdot \vec{b} = 4\). \[ |\vec{a} - \vec{b}|^2 = (2)^2 - 2(4) + (3)^2 \] \[ = 4 - 8 + 9 \] \[ = 5 \] Now, take the square root of both sides. Since magnitude must be non-negative: \[ |\vec{a} - \vec{b}| = \sqrt{5} \] Step 4: Final Answer:
The value of \(|\vec{a} - \vec{b}|\) is \(\sqrt{5}\).