Question

Which of the following is not true? If $\overrightarrow{A} = 3\hat{i} + 4\hat{j}$ and $\overrightarrow{B} = 6\hat{i} + 8\hat{j}$ where A and B are the magnitudes of $\overrightarrow{A}$ and $\overrightarrow{B}$

• A. B = 0
• B. $\frac{A}{B} = \frac{1}{2}$
• C. $\overrightarrow{A} \cdot \overrightarrow{B} = 48$
• D. A = 5

Answer 1

Answer: (A), (C)

The question asks which statement is not true. Statements (A) and (C) are not true.

Answer 2

1. Calculate the magnitude of vector $\overrightarrow{A}$: $A = |\overrightarrow{A}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
2. Calculate the magnitude of vector $\overrightarrow{B}$: $B = |\overrightarrow{B}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.
3. Calculate the dot product of $\overrightarrow{A}$ and $\overrightarrow{B}$: $\overrightarrow{A} \cdot \overrightarrow{B} = (3)(6) + (4)(8) = 18 + 32 = 50$.
4. Evaluate each option:
   • (A) $B = 0$: This is false, as $B = 10$.
   • (B) $\frac{A}{B} = \frac{1}{2}$: This is true, as $\frac{5}{10} = \frac{1}{2}$.
   • (C) $\overrightarrow{A} \cdot \overrightarrow{B} = 48$: This is false, as $\overrightarrow{A} \cdot \overrightarrow{B} = 50$.
   • (D) $A = 5$: This is true, as $A = 5$.
5. The question asks which statement is not true. The false statements are (A) and (C).