Vector Calculation: Finding Magnitude of Cross Product
The problem asks us to find the magnitude of the cross product, denoted as \(|\vec{p}\times\vec{q}|\), given specific vector definitions and properties.
We are given:
- \(\vec{p} = \vec{a} - \vec{b}\)
- \(\vec{q} = \vec{a} + \vec{b}\)
- The magnitudes of vectors \(\vec{a}\) and \(\vec{b}\) are \(|\vec{a}| = 2\) and \(|\vec{b}| = 2\).
- The dot product of \(\vec{a}\) and \(\vec{b}\) is \(\vec{a}\cdot\vec{b} = 2\).
Step 1: Calculate the Cross Product \(\vec{p} \times \vec{q}\)
First, let's find the expression for the cross product \(\vec{p} \times \vec{q}\) using the definitions of \(\vec{p}\) and \(\vec{q}\):
\(\vec{p} \times \vec{q} = (\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})\)
Using the distributive property of the cross product:
\(\vec{p} \times \vec{q} = (\vec{a} \times \vec{a}) + (\vec{a} \times \vec{b}) - (\vec{b} \times \vec{a}) - (\vec{b} \times \vec{b})\)
We use two important properties of the cross product:
- The cross product of any vector with itself is the zero vector: \(\vec{u} \times \vec{u} = \vec{0}\). So, \(\vec{a} \times \vec{a} = \vec{0}\) and \(\vec{b} \times \vec{b} = \vec{0}\).
- The cross product is anti-commutative: \(\vec{v} \times \vec{u} = -(\vec{u} \times \vec{v})\). So, \(\vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})\).
Substituting these properties back into the equation:
\(\vec{p} \times \vec{q} = \vec{0} + (\vec{a} \times \vec{b}) - (-(\vec{a} \times \vec{b})) - \vec{0}\)
\(\vec{p} \times \vec{q} = (\vec{a} \times \vec{b}) + (\vec{a} \times \vec{b})\)
\(\vec{p} \times \vec{q} = 2 (\vec{a} \times \vec{b})\)
Step 2: Calculate the Magnitude \(|\vec{p} \times \vec{q}|\)
Now, we need to find the magnitude of this resulting vector:
\(|\vec{p} \times \vec{q}| = |2 (\vec{a} \times \vec{b})|\)
Since 2 is a positive scalar, we can take it out of the magnitude:
\(|\vec{p} \times \vec{q}| = 2 |\vec{a} \times \vec{b}|\)
Step 3: Calculate the Magnitude \(|\vec{a} \times \vec{b}|\)
To find \(|\vec{a} \times \vec{b}|\), we can use the relationship between the magnitude of the cross product and the dot product:
\(|\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2\)
We are given:
- \(|\vec{a}| = 2 \implies |\vec{a}|^2 = 2^2 = 4\)
- \(|\vec{b}| = 2 \implies |\vec{b}|^2 = 2^2 = 4\)
- \(\vec{a} \cdot \vec{b} = 2\)
Substitute these values into the formula:
\(|\vec{a} \times \vec{b}|^2 = (4)(4) - (2)^2\)
\(|\vec{a} \times \vec{b}|^2 = 16 - 4\)
\(|\vec{a} \times \vec{b}|^2 = 12\)
Taking the square root to find the magnitude:
\(|\vec{a} \times \vec{b}| = \sqrt{12}\)
Simplifying the square root:
\(|\vec{a} \times \vec{b}| = \sqrt{4 \times 3} = 2\sqrt{3}\)
Step 4: Final Calculation for \(|\vec{p} \times \vec{q}|\)
Now, substitute the value of \(|\vec{a} \times \vec{b}|\) back into the expression for \(|\vec{p} \times \vec{q}|\).
\(|\vec{p} \times \vec{q}| = 2 |\vec{a} \times \vec{b}|\)
\(|\vec{p} \times \vec{q}| = 2 (2\sqrt{3})\)
\(|\vec{p} \times \vec{q}| = 4\sqrt{3}\)
Conclusion
The value of the magnitude of the cross product \(|\vec{p} \times \vec{q}|\) is \(4\sqrt{3}\). This matches the option \(4\sqrt{3}\).
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 \(\vec{a} = \hat{i} - \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} - \hat{k}\). If \(\vec{a} \times (\vec{b} \times \vec{a}) = \alpha\hat{i} - \beta\hat{j} + \gamma\hat{k}\), then what is the value of \(\alpha + \beta + \gamma\)?
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 \(2\hat{i} + \hat{j} + 2\hat{k}\)?
Important Questions from Vector Algebra
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