1. Key Concepts and Formulas

• Magnitude of a Vector: For a vector $\overrightarrow v = x\widehat i + y\widehat j + z\widehat k$, its magnitude is $|\overrightarrow v| = \sqrt{x^2 + y^2 + z^2}$. For a 2D vector $\overrightarrow v = x\widehat i + y\widehat j$, the magnitude is $|\overrightarrow v| = \sqrt{x^2 + y^2}$.
• Cross Product of Two Vectors: For vectors $\overrightarrow a = a_x\widehat i + a_y\widehat j + a_z\widehat k$ and $\overrightarrow b = b_x\widehat i + b_y\widehat j + b_z\widehat k$, their cross product is: $\overrightarrow a \times \overrightarrow b = \begin{vmatrix} \widehat i & \widehat j & \widehat k \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$

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1. Step-by-Step Solution

We are given the vectors $\overrightarrow a = 3\widehat i - 5\widehat j$ and $\overrightarrow b = 6\widehat i + 3\widehat j$. We need to find the ratio $|\overrightarrow a| : |\overrightarrow b| : |\overrightarrow c|$, where $\overrightarrow c = \overrightarrow a \times \overrightarrow b$.

Step 1: Calculate the magnitude of $\overrightarrow a$.

• Why? The ratio requires the magnitude of $\overrightarrow a$.
• Using the formula for the magnitude of a 2D vector: $|\overrightarrow a| = \sqrt{(3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}$

Step 2: Calculate the magnitude of $\overrightarrow b$.

• Why? The ratio requires the magnitude of $\overrightarrow b$.
• Using the formula for the magnitude of a 2D vector: $|\overrightarrow b| = \sqrt{(6)^2 + (3)^2} = \sqrt{36 + 9} = \sqrt{45}$

Step 3: Calculate the cross product $\overrightarrow c = \overrightarrow a \times \overrightarrow b$.

• Why? We need to find the vector $\overrightarrow c$ to calculate its magnitude. Since $\overrightarrow a$ and $\overrightarrow b$ are in the $XY$-plane, their $\widehat k$ components are zero.
• We write $\overrightarrow a = 3\widehat i - 5\widehat j + 0\widehat k$ and $\overrightarrow b = 6\widehat i + 3\widehat j + 0\widehat k$.
• Using the determinant formula for the cross product: $\overrightarrow c = \overrightarrow a \times \overrightarrow b = \begin{vmatrix} \widehat i & \widehat j & \widehat k \\ 3 & -5 & 0 \\ 6 & 3 & 0 \end{vmatrix}$
• Expanding the determinant: $\overrightarrow c = \widehat i ((-5)(0) - (0)(3)) - \widehat j ((3)(0) - (0)(6)) + \widehat k ((3)(3) - (-5)(6))$ $\overrightarrow c = \widehat i (0) - \widehat j (0) + \widehat k (9 - (-30))$ $\overrightarrow c = 0\widehat i - 0\widehat j + \widehat k (9 + 30)$ $\overrightarrow c = 39\widehat k$

Step 4: Calculate the magnitude of $\overrightarrow c$.

• Why? The ratio requires the magnitude of $\overrightarrow c$.
• The vector $\overrightarrow c$ is $0\widehat i + 0\widehat j + 39\widehat k$.
• Using the magnitude formula: $|\overrightarrow c| = \sqrt{(0)^2 + (0)^2 + (39)^2} = \sqrt{39^2} = 39$

Step 5: Form the ratio $|\overrightarrow a| : |\overrightarrow b| : |\overrightarrow c|$.

• Why? This is the final step to answer the question.
• Substituting the calculated magnitudes: $|\overrightarrow a| : |\overrightarrow b| : |\overrightarrow c| = \sqrt{34} : \sqrt{45} : 39$

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1. Common Mistakes & Tips

• Zero $\widehat k$ Component: When calculating the cross product of 2D vectors, remember to include a zero for the $\widehat k$ component of each vector. This is crucial for the determinant calculation.
• Magnitude of Cross Product: The magnitude of the cross product $|\overrightarrow a \times \overrightarrow b|$ gives the area of the parallelogram formed by $\overrightarrow a$ and $\overrightarrow b$. For vectors in the $XY$-plane, the cross product is always a vector along the $Z$-axis.
• Simplifying Square Roots: While $\sqrt{45}$ can be simplified to $3\sqrt{5}$, it's not necessary for forming the ratio as given in the options.

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1. Summary

We calculated the magnitudes of the given vectors $\overrightarrow a$ and $\overrightarrow b$. Then, we computed their cross product $\overrightarrow c$ using the determinant method, which resulted in a vector along the $Z$-axis. Finally, we found the magnitude of $\overrightarrow c$ and formed the ratio of the three magnitudes. The calculated ratio is $\sqrt{34} : \sqrt{45} : 39$.

The final answer is \boxed{\text{(B)}}.