Which of the following is not true ? If `vec(A) = 3hat(i) + 4hat(j)` and `vec(B) = 6hat(i) + 8hat(j)` where A and B are the magnitude of `vec(A)` and `vec(B)` ?

Options

A. `vec(A) xx vec(B) = 0`
B. `(|vec(A)|)/(|vec(B)|) = (1)/(2)`
C. `vec(A).vec(B) = 48`
D. `A = 5`

Answer

To solve the problem, we need to analyze the vectors \(\vec{A}\) and \(\vec{B}\) given by:

\[ \vec{A} = 3\hat{i} + 4\hat{j} \]
\[ \vec{B} = 6\hat{i} + 8\hat{j} \]

We need to determine which of the following statements is not true regarding these vectors.

### Step 1: Calculate the magnitudes of \(\vec{A}\) and \(\vec{B}\)

The magnitude of a vector \(\vec{V} = a\hat{i} + b\hat{j}\) is given by:

\[ |\vec{V}| = \sqrt{a^2 + b^2} \]

**For \(\vec{A}\):**

\[ |\vec{A}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

**For \(\vec{B}\):**

\[ |\vec{B}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

### Step 2: Calculate the ratio of the magnitudes of \(\vec{A}\) and \(\vec{B}\)

The ratio of the magnitudes is:

\[ \frac{|\vec{A}|}{|\vec{B}|} = \frac{5}{10} = \frac{1}{2} \]

### Step 3: Calculate the cross product \(\vec{A} \times \vec{B}\)

Using the determinant method for the cross product:

\[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 4 & 0 \\ 6 & 8 & 0 \end{vmatrix} \]

Calculating this determinant:

\[ = \hat{i}(4 \cdot 0 - 0 \cdot 8) - \hat{j}(3 \cdot 0 - 0 \cdot 6) + \hat{k}(3 \cdot 8 - 4 \cdot 6) \]
\[ = \hat{i}(0) - \hat{j}(0) + \hat{k}(24 - 24) \]
\[ = \hat{k}(0) = 0 \]

### Step 4: Calculate the dot product \(\vec{A} \cdot \vec{B}\)

The dot product is given by:

\[ \vec{A} \cdot \vec{B} = (3)(6) + (4)(8) = 18 + 32 = 50 \]

### Conclusion

Now we have the following results:

1. Magnitude of \(\vec{A} = 5\)
2. Magnitude of \(\vec{B} = 10\)
3. Ratio of magnitudes \(\frac{|\vec{A}|}{|\vec{B}|} = \frac{1}{2}\)
4. Cross product \(\vec{A} \times \vec{B} = 0\)
5. Dot product \(\vec{A} \cdot \vec{B} = 50\)

From the analysis, the statement regarding the dot product being equal to 48 is incorrect, as we calculated it to be 50. Therefore, the statement that is not true is related to the dot product.