© All Rights reserved @ LearnWithDash
Step 1: Understand the given information
We have three vectors \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} each of unit magnitude, i.e.,
|\overrightarrow{a}| = |\overrightarrow{b}| = |\overrightarrow{c}| = 1.
We are also given:
| \overrightarrow{a} - \overrightarrow{b} |^2 + | \overrightarrow{a} - \overrightarrow{c} |^2 = 8.
Step 2: Expand the given expressions using the dot product
Recall that for any two vectors \overrightarrow{x} and \overrightarrow{y},
| \overrightarrow{x} - \overrightarrow{y} |^2 = |\overrightarrow{x}|^2 + |\overrightarrow{y}|^2 - 2(\overrightarrow{x} \cdot \overrightarrow{y}).
Applying this to each term:
| \overrightarrow{a} - \overrightarrow{b} |^2
= |\overrightarrow{a}|^2 + |\overrightarrow{b}|^2 - 2(\overrightarrow{a} \cdot \overrightarrow{b})
= 1 + 1 - 2(\overrightarrow{a} \cdot \overrightarrow{b})
= 2 - 2(\overrightarrow{a} \cdot \overrightarrow{b}).
| \overrightarrow{a} - \overrightarrow{c} |^2
= |\overrightarrow{a}|^2 + |\overrightarrow{c}|^2 - 2(\overrightarrow{a} \cdot \overrightarrow{c})
= 1 + 1 - 2(\overrightarrow{a} \cdot \overrightarrow{c})
= 2 - 2(\overrightarrow{a} \cdot \overrightarrow{c}).
Adding these two expressions (as given):
| \overrightarrow{a} - \overrightarrow{b} |^2 + | \overrightarrow{a} - \overrightarrow{c} |^2
= \bigl(2 - 2(\overrightarrow{a} \cdot \overrightarrow{b})\bigr) + \bigl(2 - 2(\overrightarrow{a} \cdot \overrightarrow{c})\bigr).
So,
4 - 2\bigl(\overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{a} \cdot \overrightarrow{c}\bigr) = 8.
From this, we solve for \overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{a} \cdot \overrightarrow{c} :
4 - 2\bigl(\overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{a} \cdot \overrightarrow{c}\bigr) = 8 \\
\implies -2\bigl(\overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{a} \cdot \overrightarrow{c}\bigr) = 8 - 4 \\
\implies -2\bigl(\overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{a} \cdot \overrightarrow{c}\bigr) = 4 \\
\implies \overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{a} \cdot \overrightarrow{c} = -2.
Step 3: Expand the required expression
We need to find:
|\overrightarrow{a} + 2\overrightarrow{b}|^2 + |\overrightarrow{a} + 2\overrightarrow{c}|^2.
Expand using |\overrightarrow{x} + \overrightarrow{y}|^2 = |\overrightarrow{x}|^2 + |\overrightarrow{y}|^2 + 2(\overrightarrow{x} \cdot \overrightarrow{y}).
For \overrightarrow{a} + 2\overrightarrow{b},
|\overrightarrow{a} + 2\overrightarrow{b}|^2
= |\overrightarrow{a}|^2 + 4|\overrightarrow{b}|^2 + 4(\overrightarrow{a} \cdot \overrightarrow{b}).
Similarly, for \overrightarrow{a} + 2\overrightarrow{c},
|\overrightarrow{a} + 2\overrightarrow{c}|^2
= |\overrightarrow{a}|^2 + 4|\overrightarrow{c}|^2 + 4(\overrightarrow{a} \cdot \overrightarrow{c}).
Add these two:
|\overrightarrow{a} + 2\overrightarrow{b}|^2 + |\overrightarrow{a} + 2\overrightarrow{c}|^2
= (|\overrightarrow{a}|^2 + 4|\overrightarrow{b}|^2 + 4\,\overrightarrow{a} \cdot \overrightarrow{b})
+ (|\overrightarrow{a}|^2 + 4|\overrightarrow{c}|^2 + 4\,\overrightarrow{a} \cdot \overrightarrow{c}).
Step 4: Substitute the known magnitudes and dot product sum
Since |\overrightarrow{a}| = |\overrightarrow{b}| = |\overrightarrow{c}| = 1, we get
|\overrightarrow{a} + 2\overrightarrow{b}|^2 + |\overrightarrow{a} + 2\overrightarrow{c}|^2
= \bigl(1 + 4 \cdot 1 + 4(\overrightarrow{a} \cdot \overrightarrow{b})\bigr) + \bigl(1 + 4 \cdot 1 + 4(\overrightarrow{a} \cdot \overrightarrow{c})\bigr).
Combine like terms:
= (1 + 4 + 1 + 4) + 4(\overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{a} \cdot \overrightarrow{c})
= 10 + 4(\overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{a} \cdot \overrightarrow{c}).
We already found \overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{a} \cdot \overrightarrow{c} = -2.
So:
10 + 4(-2) = 10 - 8 = 2.
Step 5: State the final result
Therefore,
|\overrightarrow{a} + 2\overrightarrow{b}|^2 + |\overrightarrow{a} + 2\overrightarrow{c}|^2 = 2.
