Key Concepts and Formulas

• Vector Decomposition: Any vector $\overrightarrow b$ can be decomposed into a component $\overrightarrow {{b_1}}$ parallel to a non-zero vector $\overrightarrow a$ and a component $\overrightarrow {{b_2}}$ perpendicular to $\overrightarrow a$, such that $\overrightarrow b = \overrightarrow {{b_1}} + \overrightarrow {{b_2}}$.
• Vector Projection: The component of $\overrightarrow b$ parallel to $\overrightarrow a$ is given by the vector projection formula: $\overrightarrow {{b_1}} = \text{proj}_{\overrightarrow a} \overrightarrow b = \frac{(\overrightarrow b \cdot \overrightarrow a)}{|\overrightarrow a|^2} \overrightarrow a$
• Perpendicular Component: The component of $\overrightarrow b$ perpendicular to $\overrightarrow a$ is found by subtracting the parallel component: $\overrightarrow {{b_2}} = \overrightarrow b - \overrightarrow {{b_1}}$
• Cross Product: The cross product of two vectors $\overrightarrow u = u_1\widehat i + u_2\widehat j + u_3\widehat k$ and $\overrightarrow v = v_1\widehat i + v_2\widehat j + v_3\widehat k$ is calculated as: \widehat i & \widehat j & \widehat k \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$$

Step-by-Step Solution

Step 1: Understand the Problem and Identify Given Vectors Why: To begin any vector problem, it's crucial to clearly identify the vectors involved and what is being asked. We are given:

• $\overrightarrow b = 3\widehat j + 4\widehat k$
• $\overrightarrow a = \widehat i + \widehat j$ We need to find $\overrightarrow {{b_1}} \times \overrightarrow {{b_2}}$, where $\overrightarrow {{b_1}}$ is parallel to $\overrightarrow a$ and $\overrightarrow {{b_2}}$ is perpendicular to $\overrightarrow a$, and $\overrightarrow b = \overrightarrow {{b_1}} + \overrightarrow {{b_2}}$.

Step 2: Calculate the Dot Product $\overrightarrow b \cdot \overrightarrow a$ Why: The dot product is a fundamental operation used in the vector projection formula to determine how much of one vector lies in the direction of another. $\overrightarrow b \cdot \overrightarrow a = (0\widehat i + 3\widehat j + 4\widehat k) \cdot (\widehat i + \widehat j + 0\widehat k)$ $= (0)(1) + (3)(1) + (4)(0) = 0 + 3 + 0 = 3$

Step 3: Calculate the Square of the Magnitude of $\overrightarrow a$, $|\overrightarrow a|^2$ Why: The squared magnitude of the vector $\overrightarrow a$ is needed as the denominator in the vector projection formula. Calculating the square avoids dealing with square roots prematurely, simplifying the calculation. $|\overrightarrow a|^2 = (\widehat i + \widehat j) \cdot (\widehat i + \widehat j) = (1)^2 + (1)^2 + (0)^2 = 1 + 1 + 0 = 2$

Step 4: Determine the Vector Component $\overrightarrow {{b_1}}$ Parallel to $\overrightarrow a$ Why: This step applies the vector projection formula to find the part of $\overrightarrow b$ that aligns with the direction of $\overrightarrow a$. Using the results from Step 2 and Step 3: $\overrightarrow {{b_1}} = \frac{\overrightarrow b \cdot \overrightarrow a}{|\overrightarrow a|^2} \overrightarrow a = \frac{3}{2} (\widehat i + \widehat j)$ $\overrightarrow {{b_1}} = \frac{3}{2}\widehat i + \frac{3}{2}\widehat j$

Step 5: Determine the Vector Component $\overrightarrow {{b_2}}$ Perpendicular to $\overrightarrow a$ Why: Since $\overrightarrow b$ is the sum of its parallel and perpendicular components, we can find the perpendicular component by subtracting the parallel component from the original vector. $\overrightarrow {{b_2}} = \overrightarrow b - \overrightarrow {{b_1}}$ Substituting the given $\overrightarrow b$ and the calculated $\overrightarrow {{b_1}}$: $\overrightarrow {{b_2}} = (0\widehat i + 3\widehat j + 4\widehat k) - \left(\frac{3}{2}\widehat i + \frac{3}{2}\widehat j + 0\widehat k\right)$ Combining the components: $= \left(0 - \frac{3}{2}\right)\widehat i + \left(3 - \frac{3}{2}\right)\widehat j + (4 - 0)\widehat k$ $= -\frac{3}{2}\widehat i + \left(\frac{6-3}{2}\right)\widehat j + 4\widehat k$ $\overrightarrow {{b_2}} = -\frac{3}{2}\widehat i + \frac{3}{2}\widehat j + 4\widehat k$

Step 6: Verify Perpendicularity (Optional but Recommended) Why: To ensure our calculations for $\overrightarrow {{b_2}}$ are correct, we can check if it is indeed perpendicular to $\overrightarrow a$ by calculating their dot product. If the dot product is zero, the vectors are perpendicular. $\overrightarrow {{b_2}} \cdot \overrightarrow a = \left(-\frac{3}{2}\widehat i + \frac{3}{2}\widehat j + 4\widehat k\right) \cdot (\widehat i + \widehat j + 0\widehat k)$ $= \left(-\frac{3}{2}\right)(1) + \left(\frac{3}{2}\right)(1) + (4)(0) = -\frac{3}{2} + \frac{3}{2} + 0 = 0$ The dot product is 0, confirming that $\overrightarrow {{b_2}}$ is perpendicular to $\overrightarrow a$.

Step 7: Calculate the Cross Product $\overrightarrow {{b_1}} \times \overrightarrow {{b_2}}$ Why: This is the final calculation required by the problem. The cross product will yield a vector perpendicular to both $\overrightarrow {{b_1}}$ and $\overrightarrow {{b_2}}$. We have:

• $\overrightarrow {{b_1}} = \frac{3}{2}\widehat i + \frac{3}{2}\widehat j + 0\widehat k$
• $\overrightarrow {{b_2}} = -\frac{3}{2}\widehat i + \frac{3}{2}\widehat j + 4\widehat k$

Using the determinant formula for the cross product:

\widehat i & \widehat j & \widehat k \\ \frac{3}{2} & \frac{3}{2} & 0 \\ -\frac{3}{2} & \frac{3}{2} & 4 \end{vmatrix}$$ Expanding the determinant: $$ = \widehat i \left( \left(\frac{3}{2}\right)(4) - (0)\left(\frac{3}{2}\right) \right) - \widehat j \left( \left(\frac{3}{2}\right)(4) - (0)\left(-\frac{3}{2}\right) \right) + \widehat k \left( \left(\frac{3}{2}\right)\left(\frac{3}{2}\right) - \left(\frac{3}{2}\right)\left(-\frac{3}{2}\right) \right)$$ $$ = \widehat i (6 - 0) - \widehat j (6 - 0) + \widehat k \left(\frac{9}{4} - \left(-\frac{9}{4}\right)\right)$$ $$ = 6\widehat i - 6\widehat j + \widehat k \left(\frac{9}{4} + \frac{9}{4}\right)$$ $$ = 6\widehat i - 6\widehat j + \widehat k \left(\frac{18}{4}\right)$$ $$ = 6\widehat i - 6\widehat j + \frac{9}{2}\widehat k$$ **Common Mistakes & Tips** * **Sign Errors in Cross Product:** Be extremely careful with the signs when calculating the determinant, especially for the $\widehat j$ component, which has a negative sign in the expansion. * **Misinterpreting Projection:** Distinguish between scalar projection (a length) and vector projection (a vector). The formula for vector projection requires the vector $\overrightarrow a$, not just its magnitude. * **Algebraic Slip-ups:** Double-check all arithmetic, especially when dealing with fractions, to avoid calculation errors. **Summary** The problem requires decomposing a vector $\overrightarrow b$ into components parallel and perpendicular to another vector $\overrightarrow a$. This is achieved using the vector projection formula for the parallel component ($\overrightarrow {{b_1}}$) and then subtracting this from $\overrightarrow b$ to find the perpendicular component ($\overrightarrow {{b_2}}$). Finally, the cross product of $\overrightarrow {{b_1}}$ and $\overrightarrow {{b_2}}$ is computed. The steps involved calculating dot products, magnitudes, performing vector subtraction, and evaluating a determinant for the cross product. The final answer is $6\widehat i - 6\widehat j + \frac{9}{2}\widehat k$, which corresponds to option (B). The final answer is $\boxed{6\widehat i - 6\widehat j + {9 \over 2}\widehat k}$.