Question:
Which of the following is not true about vectors \(\vec{A},\vec{B}\) and \(\vec{C}\)?
Which of the following is not true about vectors \(\vec{A},\vec{B}\) and \(\vec{C}\)?
Show Hint
Remember:
\[
\text{Dot product} \Rightarrow \text{Scalar}
\]
\[
\text{Cross product} \Rightarrow \text{Vector}
\]
This helps quickly identify the nature of vector expressions.
Updated On: Jun 22, 2026
- \((\vec{A}\cdot\vec{A})(\vec{B}\cdot\vec{C})\) is a scalar value.
- \((\vec{A}\times\vec{B})\cdot(\vec{B}\times\vec{C})\) is a scalar value.
- \((\vec{A}\times\vec{C})\times(\vec{B}\times\vec{C})\) is a scalar value.
- \(\vec{A}\times(\vec{B}\times\vec{C})\) is a vector value.
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The Correct Option is C
Solution and Explanation
Step 1: Check option (1).
\[ (\vec{A}\cdot\vec{A}) \] is a dot product, so it is a scalar quantity.
Also, \[ (\vec{B}\cdot\vec{C}) \] is a scalar quantity.
Product of two scalars is again a scalar.
Hence, option (1) is true.
Step 2: Check option (2).
\[ (\vec{A}\times\vec{B}) \] and \[ (\vec{B}\times\vec{C}) \] are vectors.
Their dot product \[ (\vec{A}\times\vec{B})\cdot(\vec{B}\times\vec{C}) \] is therefore a scalar quantity.
Hence, option (2) is true.
Step 3: Check option (3).
\[ (\vec{A}\times\vec{C}) \] and \[ (\vec{B}\times\vec{C}) \] are vectors.
Now consider \[ (\vec{A}\times\vec{C})\times(\vec{B}\times\vec{C}) \] This is a cross product of two vectors.
A cross product always gives a vector quantity, not a scalar quantity.
Therefore, the statement saying it is a scalar value is false.
Hence, option (3) is not true.
Step 4: Check option (4).
\[ \vec{B}\times\vec{C} \] is a vector.
Then, \[ \vec{A}\times(\vec{B}\times\vec{C}) \] is again a cross product of vectors, which gives a vector quantity.
Hence, option (4) is true.
Step 5: Final conclusion.
Thus, the incorrect statement is \[ \boxed{(\vec{A}\times\vec{C})\times(\vec{B}\times\vec{C})\text{ is a scalar value}} \]
\[ (\vec{A}\cdot\vec{A}) \] is a dot product, so it is a scalar quantity.
Also, \[ (\vec{B}\cdot\vec{C}) \] is a scalar quantity.
Product of two scalars is again a scalar.
Hence, option (1) is true.
Step 2: Check option (2).
\[ (\vec{A}\times\vec{B}) \] and \[ (\vec{B}\times\vec{C}) \] are vectors.
Their dot product \[ (\vec{A}\times\vec{B})\cdot(\vec{B}\times\vec{C}) \] is therefore a scalar quantity.
Hence, option (2) is true.
Step 3: Check option (3).
\[ (\vec{A}\times\vec{C}) \] and \[ (\vec{B}\times\vec{C}) \] are vectors.
Now consider \[ (\vec{A}\times\vec{C})\times(\vec{B}\times\vec{C}) \] This is a cross product of two vectors.
A cross product always gives a vector quantity, not a scalar quantity.
Therefore, the statement saying it is a scalar value is false.
Hence, option (3) is not true.
Step 4: Check option (4).
\[ \vec{B}\times\vec{C} \] is a vector.
Then, \[ \vec{A}\times(\vec{B}\times\vec{C}) \] is again a cross product of vectors, which gives a vector quantity.
Hence, option (4) is true.
Step 5: Final conclusion.
Thus, the incorrect statement is \[ \boxed{(\vec{A}\times\vec{C})\times(\vec{B}\times\vec{C})\text{ is a scalar value}} \]
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