Vector Triple Product Calculation
This solution explains how to find the value of \(\alpha + \beta + \gamma\) given two vectors \(\vec{a} = \hat{i} - \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} - \hat{k}\), and the condition \(\vec{a} \times (\vec{b} \times \vec{a}) = \alpha\hat{i} - \beta\hat{j} + \gamma\hat{k}\).
Understanding the Vector Triple Product
The problem involves calculating a vector triple product, which is the cross product of three vectors. A standard identity simplifies this calculation:
\(\vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w}\)For this problem, we set \(\vec{u} = \vec{a}\), \(\vec{v} = \vec{b}\), and \(\vec{w} = \vec{a}\). Applying this to the identity gives:
\(\vec{a} \times (\vec{b} \times \vec{a}) = (\vec{a} \cdot \vec{a})\vec{b} - (\vec{a} \cdot \vec{b})\vec{a}\)Step 1: Calculate Necessary Dot Products
Before applying the identity, we need to calculate two dot products:
- Calculate \(\vec{a} \cdot \vec{a}\):
Given \(\vec{a} = \hat{i} - \hat{j} + \hat{k}\). The dot product \(\vec{a} \cdot \vec{a}\) is the square of the magnitude of \(\vec{a}\). \(\vec{a} \cdot \vec{a} = (1)(1) + (-1)(-1) + (1)(1) = 1 + 1 + 1 = 3\) - Calculate \(\vec{a} \cdot \vec{b}\):
Given \(\vec{b} = \hat{i} + 2\hat{j} - \hat{k}\). \(\vec{a} \cdot \vec{b} = (\hat{i} - \hat{j} + \hat{k}) \cdot (\hat{i} + 2\hat{j} - \hat{k})\) \(\vec{a} \cdot \vec{b} = (1)(1) + (-1)(2) + (1)(-1) = 1 - 2 - 1 = -2\)
Step 2: Use the Identity to Find the Resultant Vector
Substitute the computed dot product values back into the vector triple product identity:
\(\vec{a} \times (\vec{b} \times \vec{a}) = (3)\vec{b} - (-2)\vec{a}\)This simplifies to:
\(\vec{a} \times (\vec{b} \times \vec{a}) = 3\vec{b} + 2\vec{a}\)Step 3: Express the Result in terms of \(\hat{i}, \hat{j}, \hat{k}\)
Now, substitute the component forms of \(\vec{a}\) and \(\vec{b}\):
- Calculate \(3\vec{b}\): \(3\vec{b} = 3(\hat{i} + 2\hat{j} - \hat{k}) = 3\hat{i} + 6\hat{j} - 3\hat{k}\)
- Calculate \(2\vec{a}\): \(2\vec{a} = 2(\hat{i} - \hat{j} + \hat{k}) = 2\hat{i} - 2\hat{j} + 2\hat{k}\)
Add the results of \(3\vec{b}\) and \(2\vec{a}\):
\(3\vec{b} + 2\vec{a} = (3\hat{i} + 6\hat{j} - 3\hat{k}) + (2\hat{i} - 2\hat{j} + 2\hat{k})\)Combine the components:
\(3\vec{b} + 2\vec{a} = (3+2)\hat{i} + (6-2)\hat{j} + (-3+2)\hat{k}\) \(3\vec{b} + 2\vec{a} = 5\hat{i} + 4\hat{j} - \hat{k}\)Thus, the vector resulting from \(\vec{a} \times (\vec{b} \times \vec{a})\) is \(5\hat{i} + 4\hat{j} - \hat{k}\).
Step 4: Identify \(\alpha\), \(\beta\), and \(\gamma\)
We are given that the result equals \(\alpha\hat{i} - \beta\hat{j} + \gamma\hat{k}\). By comparing this with our calculated vector \(5\hat{i} + 4\hat{j} - \hat{k}\), we can determine the values:
\(\alpha\hat{i} - \beta\hat{j} + \gamma\hat{k} = 5\hat{i} + 4\hat{j} - \hat{k}\)- Comparing the coefficients of \(\hat{i}\): \(\alpha = 5\)
- Comparing the coefficients of \(\hat{j}\): \(-\beta = 4 \implies \beta = -4\)
- Comparing the coefficients of \(\hat{k}\): \(\gamma = -1\)
Step 5: Compute the Sum \(\alpha + \beta + \gamma\)
Finally, calculate the sum using the determined values of \(\alpha\), \(\beta\), and \(\gamma\):
\(\alpha + \beta + \gamma = 5 + (-4) + (-1)\) \(\alpha + \beta + \gamma = 5 - 4 - 1\) \(\alpha + \beta + \gamma = 0\)The value of the sum \(\alpha + \beta + \gamma\) is 0.
Similar Questions
Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-zero vectors such that \(\vec{a}\times \vec{b} = \vec{c} \) . Consider the following statements:
1. \(\vec a\) is unique if \(\vec b\) and \(\vec c\) are given
2. \(\vec c\) is unique if \(\vec a\) and \(\vec b\) are given
Which of the above statements is/are correct?
In a right angled triangle ABC, if the hypotenuse AC = p, then what is \(\overrightarrow {{\rm{AB}}} \cdot \overrightarrow {{\rm{AC}}} + \overrightarrow {{\rm{BC}}} \cdot \overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{CA}}} \cdot \overrightarrow {{\rm{CB}}} \) equal to?
What is \({\rm{\vec c}}\) equal to?
If \({\rm{\vec d}} = {\rm{x\hat i}} + {\rm{y\hat j}} + {\rm{z\hat k}}\) , then which of the following equations is/are correct?
1. y – x = 4
2. 2z – 3 = 0
Select the correct answer using the code given below:
What is \({\rm{\vec a}} \cdot {\rm{\vec b}} + {\rm{\vec b}} \cdot {\rm{\vec c}} + {\rm{\vec c}} \cdot {\rm{\vec a}}\) equal to?
What is the angle between \({\rm{\vec a}}\) and \({\rm{\vec b}}\) ?
- What is the fourth vertex \(D\)?
- Let \(\vec{p}=\vec{a}-\vec{b}\), \(\vec{q}=\vec{a}+\vec{b}\). If \(|\vec{a}|=|\vec{b}|= 2\) and \(\vec{a}\cdot\vec{b} = 2\), then what is the value of \(|\vec{p}\times\vec{q}|\)?
How many of the following can be a vector perpendicular to both the vectors \(2\hat{i} - \hat{j} + \hat{k}\) and \(\hat{i} + \hat{j} + 3\hat{k}\) ?
I. \(4\hat{i} + 5\hat{j} - 3\hat{k}\)
II. \(-8\hat{i} - 10\hat{j} + 6\hat{k}\)
III. \(\frac{1}{50}(-4\hat{i}-5\hat{j}+3\hat{k})\)
Select the correct answer.
- What is the area of the parallelogram whose sides are represented by the vectors \(\hat{i} + 2\hat{j} + 3\hat{k}\) and \(2\hat{i} + \hat{j} + 2\hat{k}\)?
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