Two adjacent sides of a parallelogram are 2î - 4ĵ + 5k̂ and î - 2ĵ - 3k̂. What is the magnitude of dot product of vectors which represent its diagonals?
31
Understanding Parallelogram Vectors and Diagonals
This problem involves finding the magnitude of the dot product between the vectors representing the diagonals of a parallelogram. We are given two adjacent sides of the parallelogram as vectors. Let these vectors be denoted by \(\vec{a}\) and \(\vec{b}\).
Given:
- Vector representing one adjacent side: \(\vec{a} = 2\hat{i} - 4\hat{j} + 5\hat{k}\)
- Vector representing the other adjacent side: \(\vec{b} = \hat{i} - 2\hat{j} - 3\hat{k}\)
In a parallelogram, the vectors representing the diagonals can be found by the sum and difference of the vectors representing the adjacent sides.
- Let the first diagonal vector be \(\vec{d_1}\).
- Let the second diagonal vector be \(\vec{d_2}\).
Calculating the Two Diagonal Vectors
The diagonal vectors are calculated as follows:
- First Diagonal (\(\vec{d_1}\)): This is the sum of the two adjacent side vectors.
\(\vec{d_1} = \vec{a} + \vec{b}\)
Substitute the given vectors:
\(\vec{d_1} = (2\hat{i} - 4\hat{j} + 5\hat{k}) + (\hat{i} - 2\hat{j} - 3\hat{k})\)
Combine the components:
\(\vec{d_1} = (2+1)\hat{i} + (-4-2)\hat{j} + (5-3)\hat{k}\)
\(\vec{d_1} = 3\hat{i} - 6\hat{j} + 2\hat{k}\)
- Second Diagonal (\(\vec{d_2}\)): This is the difference between the two adjacent side vectors.
\(\vec{d_2} = \vec{a} - \vec{b}\)
Substitute the given vectors:
\(\vec{d_2} = (2\hat{i} - 4\hat{j} + 5\hat{k}) - (\hat{i} - 2\hat{j} - 3\hat{k})\)
Distribute the negative sign and combine the components:
\(\vec{d_2} = (2-1)\hat{i} + (-4 - (-2))\hat{j} + (5 - (-3))\hat{k}\)
\(\vec{d_2} = (2-1)\hat{i} + (-4+2)\hat{j} + (5+3)\hat{k}\)
\(\vec{d_2} = 1\hat{i} - 2\hat{j} + 8\hat{k}\)
Computing the Dot Product of the Diagonal Vectors
Now, we calculate the dot product of the two diagonal vectors, \(\vec{d_1}\) and \(\vec{d_2}\). The dot product of two vectors \(\vec{p} = p_1\hat{i} + p_2\hat{j} + p_3\hat{k}\) and \(\vec{q} = q_1\hat{i} + q_2\hat{j} + q_3\hat{k}\) is given by \(\vec{p} \cdot \vec{q} = p_1q_1 + p_2q_2 + p_3q_3\).
\(\vec{d_1} \cdot \vec{d_2} = (3\hat{i} - 6\hat{j} + 2\hat{k}) \cdot (1\hat{i} - 2\hat{j} + 8\hat{k})\)
Multiply corresponding components and sum them:
\(\vec{d_1} \cdot \vec{d_2} = (3)(1) + (-6)(-2) + (2)(8)\)
\(\vec{d_1} \cdot \vec{d_2} = 3 + 12 + 16\)
\(\vec{d_1} \cdot \vec{d_2} = 31\)
Finding the Magnitude of the Dot Product
The question asks for the magnitude of the dot product of the vectors representing its diagonals. The dot product we calculated is a scalar value, 31.
The magnitude of a scalar value is its absolute value.
Magnitude = \(| \vec{d_1} \cdot \vec{d_2} | = |31| = 31\).
Final Result
The magnitude of the dot product of the vectors representing the diagonals of the parallelogram is 31.
Comparing this result with the given options:
- Option 1: 21
- Option 2: 25
- Option 3: 31
- Option 4: 36
The calculated value matches Option 3.
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