Question

If \(|\bar a|=10\ and\ |\bar b|=5\), then the value of \((\bar a+2\bar b)\cdot(\bar a-2\bar b)\) is equal to

• A. 32
• B. 16
• C. 8
• D. 4
• E. 0

Answer 1

Answer: (E)

We are asked to find the value of \( (\vec{a} + 2\vec{b}) \cdot (\vec{a} - 2\vec{b}) \). We can expand the dot product as follows: \[ (\vec{a} + 2\vec{b}) \cdot (\vec{a} - 2\vec{b}) = \vec{a} \cdot \vec{a} - 2\vec{a} \cdot \vec{b} + 2\vec{b} \cdot \vec{a} - 4\vec{b} \cdot \vec{b} \] We can simplify this expression using the properties of dot products: \[ = \vec{a} \cdot \vec{a} - 4 \vec{b} \cdot \vec{b} \] Note that \( \vec{a} \cdot \vec{a} = |\vec{a}|^2 \) and \( \vec{b} \cdot \vec{b} = |\vec{b}|^2 \). Therefore, we get: \[ = |\vec{a}|^2 - 4|\vec{b}|^2 \] Substitute the given values \( |\vec{a}| = 10 \) and \( |\vec{b}| = 5 \): \[ = 10^2 - 4(5^2) = 100 - 4(25) = 100 - 100 = 0 \] Thus, the value of \( (\vec{a} + 2\vec{b}) \cdot (\vec{a} - 2\vec{b}) \) is \( {0} \).

The correct option is (E) : \(0\)

Answer 2

We are given that \(|\bar{a}| = 10\) and \(|\bar{b}| = 5\). We want to find the value of \((\bar{a} + 2\bar{b})\cdot(\bar{a} - 2\bar{b})\).

Using the distributive property of the dot product, we have:

\((\bar{a} + 2\bar{b})\cdot(\bar{a} - 2\bar{b}) = \bar{a}\cdot\bar{a} - 2\bar{a}\cdot\bar{b} + 2\bar{b}\cdot\bar{a} - 4\bar{b}\cdot\bar{b}\)

Since the dot product is commutative, \(\bar{a}\cdot\bar{b} = \bar{b}\cdot\bar{a}\), so the middle two terms cancel out:

\((\bar{a} + 2\bar{b})\cdot(\bar{a} - 2\bar{b}) = \bar{a}\cdot\bar{a} - 4\bar{b}\cdot\bar{b} = |\bar{a}|^2 - 4|\bar{b}|^2\)

Substituting the given values, we have:

\(|\bar{a}|^2 - 4|\bar{b}|^2 = (10)^2 - 4(5)^2 = 100 - 4(25) = 100 - 100 = 0\)

Therefore, the value of \((\bar{a} + 2\bar{b})\cdot(\bar{a} - 2\bar{b})\) is 0.