355555
If a vector \(2 \hat{i}+3 \hat{j}+8 \hat{k}\) is perpendicular to the vector \(4 \hat{j}-4 \hat{i}+\alpha \hat{k}\). Then the value of \(\alpha\) is
1 -1
2 \(1 / 2\)
3 \(-1 / 2\)
4 1
Explanation:
Let the given vectors be \(\vec{A}=2 \hat{i}+3 \hat{j}+8 \hat{k} \text { and } \vec{B}=-4 \hat{i}+4 \hat{j}+\alpha \hat{k}\) Dot product of these vectors should be equal to zero because they are perpendicular. \(\therefore \vec{A} \cdot \vec{B}=-8+12+8 \alpha=0 \Rightarrow 8 \alpha=-4 \Rightarrow \alpha=-1 / 2\)
PHXI06:WORK ENERGY AND POWER
355556
If two vectors \(\vec{A}\) and \(\vec{B}\) having equal magnitude \(R\) are inclined at an angle \(\theta\), then
1 \(|\vec{A}-\vec{B}|=\sqrt{2} R \sin \left(\dfrac{\theta}{2}\right)\)
2 \(|\vec{A}+\vec{B}|=2 R \sin \left(\dfrac{\theta}{2}\right)\)
3 \(|\vec{A}+\vec{B}|=2 R \cos \left(\dfrac{\theta}{2}\right)\)
4 \(|\vec{A}-\vec{B}|=2 R \cos \left(\dfrac{\theta}{2}\right)\)
355557
Assertion : \(\vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A}\) Reason : Dot product of two vectors is commutative.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Option (1) is correct
PHXI06:WORK ENERGY AND POWER
355558
The condition under which vectors \((a+b)\) and (\(a-b\) ) should be at right angles to each other is
1 \(a \neq b\)
2 \(a \cdot b=0\)
3 \(|a|=|b|\)
4 \(a \cdot b=1\)
Explanation:
The dot product of two vectors should be equal to zero, i.e. \((a + b) \cdot (a - b) = 0\) \( \Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,{a^2} - {b^2} = 0\) \(\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,|a| = |b|\)
355555
If a vector \(2 \hat{i}+3 \hat{j}+8 \hat{k}\) is perpendicular to the vector \(4 \hat{j}-4 \hat{i}+\alpha \hat{k}\). Then the value of \(\alpha\) is
1 -1
2 \(1 / 2\)
3 \(-1 / 2\)
4 1
Explanation:
Let the given vectors be \(\vec{A}=2 \hat{i}+3 \hat{j}+8 \hat{k} \text { and } \vec{B}=-4 \hat{i}+4 \hat{j}+\alpha \hat{k}\) Dot product of these vectors should be equal to zero because they are perpendicular. \(\therefore \vec{A} \cdot \vec{B}=-8+12+8 \alpha=0 \Rightarrow 8 \alpha=-4 \Rightarrow \alpha=-1 / 2\)
PHXI06:WORK ENERGY AND POWER
355556
If two vectors \(\vec{A}\) and \(\vec{B}\) having equal magnitude \(R\) are inclined at an angle \(\theta\), then
1 \(|\vec{A}-\vec{B}|=\sqrt{2} R \sin \left(\dfrac{\theta}{2}\right)\)
2 \(|\vec{A}+\vec{B}|=2 R \sin \left(\dfrac{\theta}{2}\right)\)
3 \(|\vec{A}+\vec{B}|=2 R \cos \left(\dfrac{\theta}{2}\right)\)
4 \(|\vec{A}-\vec{B}|=2 R \cos \left(\dfrac{\theta}{2}\right)\)
355557
Assertion : \(\vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A}\) Reason : Dot product of two vectors is commutative.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Option (1) is correct
PHXI06:WORK ENERGY AND POWER
355558
The condition under which vectors \((a+b)\) and (\(a-b\) ) should be at right angles to each other is
1 \(a \neq b\)
2 \(a \cdot b=0\)
3 \(|a|=|b|\)
4 \(a \cdot b=1\)
Explanation:
The dot product of two vectors should be equal to zero, i.e. \((a + b) \cdot (a - b) = 0\) \( \Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,{a^2} - {b^2} = 0\) \(\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,|a| = |b|\)
355555
If a vector \(2 \hat{i}+3 \hat{j}+8 \hat{k}\) is perpendicular to the vector \(4 \hat{j}-4 \hat{i}+\alpha \hat{k}\). Then the value of \(\alpha\) is
1 -1
2 \(1 / 2\)
3 \(-1 / 2\)
4 1
Explanation:
Let the given vectors be \(\vec{A}=2 \hat{i}+3 \hat{j}+8 \hat{k} \text { and } \vec{B}=-4 \hat{i}+4 \hat{j}+\alpha \hat{k}\) Dot product of these vectors should be equal to zero because they are perpendicular. \(\therefore \vec{A} \cdot \vec{B}=-8+12+8 \alpha=0 \Rightarrow 8 \alpha=-4 \Rightarrow \alpha=-1 / 2\)
PHXI06:WORK ENERGY AND POWER
355556
If two vectors \(\vec{A}\) and \(\vec{B}\) having equal magnitude \(R\) are inclined at an angle \(\theta\), then
1 \(|\vec{A}-\vec{B}|=\sqrt{2} R \sin \left(\dfrac{\theta}{2}\right)\)
2 \(|\vec{A}+\vec{B}|=2 R \sin \left(\dfrac{\theta}{2}\right)\)
3 \(|\vec{A}+\vec{B}|=2 R \cos \left(\dfrac{\theta}{2}\right)\)
4 \(|\vec{A}-\vec{B}|=2 R \cos \left(\dfrac{\theta}{2}\right)\)
355557
Assertion : \(\vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A}\) Reason : Dot product of two vectors is commutative.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Option (1) is correct
PHXI06:WORK ENERGY AND POWER
355558
The condition under which vectors \((a+b)\) and (\(a-b\) ) should be at right angles to each other is
1 \(a \neq b\)
2 \(a \cdot b=0\)
3 \(|a|=|b|\)
4 \(a \cdot b=1\)
Explanation:
The dot product of two vectors should be equal to zero, i.e. \((a + b) \cdot (a - b) = 0\) \( \Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,{a^2} - {b^2} = 0\) \(\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,|a| = |b|\)
355555
If a vector \(2 \hat{i}+3 \hat{j}+8 \hat{k}\) is perpendicular to the vector \(4 \hat{j}-4 \hat{i}+\alpha \hat{k}\). Then the value of \(\alpha\) is
1 -1
2 \(1 / 2\)
3 \(-1 / 2\)
4 1
Explanation:
Let the given vectors be \(\vec{A}=2 \hat{i}+3 \hat{j}+8 \hat{k} \text { and } \vec{B}=-4 \hat{i}+4 \hat{j}+\alpha \hat{k}\) Dot product of these vectors should be equal to zero because they are perpendicular. \(\therefore \vec{A} \cdot \vec{B}=-8+12+8 \alpha=0 \Rightarrow 8 \alpha=-4 \Rightarrow \alpha=-1 / 2\)
PHXI06:WORK ENERGY AND POWER
355556
If two vectors \(\vec{A}\) and \(\vec{B}\) having equal magnitude \(R\) are inclined at an angle \(\theta\), then
1 \(|\vec{A}-\vec{B}|=\sqrt{2} R \sin \left(\dfrac{\theta}{2}\right)\)
2 \(|\vec{A}+\vec{B}|=2 R \sin \left(\dfrac{\theta}{2}\right)\)
3 \(|\vec{A}+\vec{B}|=2 R \cos \left(\dfrac{\theta}{2}\right)\)
4 \(|\vec{A}-\vec{B}|=2 R \cos \left(\dfrac{\theta}{2}\right)\)
355557
Assertion : \(\vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A}\) Reason : Dot product of two vectors is commutative.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
Option (1) is correct
PHXI06:WORK ENERGY AND POWER
355558
The condition under which vectors \((a+b)\) and (\(a-b\) ) should be at right angles to each other is
1 \(a \neq b\)
2 \(a \cdot b=0\)
3 \(|a|=|b|\)
4 \(a \cdot b=1\)
Explanation:
The dot product of two vectors should be equal to zero, i.e. \((a + b) \cdot (a - b) = 0\) \( \Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,{a^2} - {b^2} = 0\) \(\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,|a| = |b|\)
