Key Concepts and Formulas

• Vector Decomposition: Any vector $\overrightarrow \beta$ can be uniquely decomposed into a component $\overrightarrow \beta_1$ parallel to a non-zero vector $\overrightarrow \alpha$ and a component $\overrightarrow \beta_2$ perpendicular to $\overrightarrow \alpha$, such that $\overrightarrow \beta = \overrightarrow \beta_1 + \overrightarrow \beta_2$.
• Vector Projection: The component of $\overrightarrow \beta$ parallel to $\overrightarrow \alpha$ is given by the vector projection of $\overrightarrow \beta$ onto $\overrightarrow \alpha$: $\overrightarrow \beta_1 = \text{proj}_{\overrightarrow \alpha} \overrightarrow \beta = \left( \frac{\overrightarrow \beta \cdot \overrightarrow \alpha}{\left\| \overrightarrow \alpha \right\|^2} \right) \overrightarrow \alpha$
• Dot Product: For vectors $\overrightarrow{u} = x_1\widehat i + y_1\widehat j + z_1\widehat k$ and $\overrightarrow{v} = x_2\widehat i + y_2\widehat j + z_2\widehat k$, their dot product is $\overrightarrow{u} \cdot \overrightarrow{v} = x_1x_2 + y_1y_2 + z_1z_2$.
• Magnitude Squared: For a vector $\overrightarrow{u} = x\widehat i + y\widehat j + z\widehat k$, its magnitude squared is $\|\overrightarrow{u}\|^2 = x^2 + y^2 + z^2$.
• Perpendicular Vectors: Two non-zero vectors are perpendicular if their dot product is zero.

Step-by-Step Solution

Step 1: Understand the Problem and Given Information We are given two vectors, $\overrightarrow \alpha = 4\widehat i + 3\widehat j + 5\widehat k$ and $\overrightarrow \beta = \widehat i + 2\widehat j - 4\widehat k$. We are told that $\overrightarrow \beta$ is decomposed into two components: $\overrightarrow \beta_1$ parallel to $\overrightarrow \alpha$, and $\overrightarrow \beta_2$ perpendicular to $\overrightarrow \alpha$, such that $\overrightarrow \beta = \overrightarrow \beta_1 + \overrightarrow \beta_2$. Our goal is to find the value of $5{\overrightarrow \beta _2}\,.\left( {\widehat i + \widehat j + \widehat k} \right)$.

Step 2: Calculate the Component Parallel to $\overrightarrow \alpha$ ($\overrightarrow \beta_1$) To find $\overrightarrow \beta_1$, we use the formula for vector projection. First, we need to compute the dot product of $\overrightarrow \beta$ and $\overrightarrow \alpha$, and the squared magnitude of $\overrightarrow \alpha$.

Calculate $\overrightarrow \beta \cdot \overrightarrow \alpha$: $\overrightarrow \beta \cdot \overrightarrow \alpha = (\widehat i + 2\widehat j - 4\widehat k) \cdot (4\widehat i + 3\widehat j + 5\widehat k)$ $= (1)(4) + (2)(3) + (-4)(5)$ $= 4 + 6 - 20$ $= -10$

Calculate $\|\overrightarrow \alpha\|^2$: $\|\overrightarrow \alpha\|^2 = (4)^2 + (3)^2 + (5)^2$ $= 16 + 9 + 25$ $= 50$

Now, substitute these values into the vector projection formula for $\overrightarrow \beta_1$: $\overrightarrow \beta_1 = \left( \frac{\overrightarrow \beta \cdot \overrightarrow \alpha}{\left\| \overrightarrow \alpha \right\|^2} \right) \overrightarrow \alpha$ $\overrightarrow \beta_1 = \left( \frac{-10}{50} \right) (4\widehat i + 3\widehat j + 5\widehat k)$ $\overrightarrow \beta_1 = -\frac{1}{5} (4\widehat i + 3\widehat j + 5\widehat k)$ $\overrightarrow \beta_1 = -\frac{4}{5}\widehat i - \frac{3}{5}\widehat j - \widehat k$

Step 3: Calculate the Component Perpendicular to $\overrightarrow \alpha$ ($\overrightarrow \beta_2$) Since $\overrightarrow \beta = \overrightarrow \beta_1 + \overrightarrow \beta_2$, we can find $\overrightarrow \beta_2$ by subtracting $\overrightarrow \beta_1$ from $\overrightarrow \beta$: $\overrightarrow \beta_2 = \overrightarrow \beta - \overrightarrow \beta_1$ $\overrightarrow \beta_2 = (\widehat i + 2\widehat j - 4\widehat k) - \left( -\frac{4}{5}\widehat i - \frac{3}{5}\widehat j - \widehat k \right)$ Distribute the negative sign and combine like terms: $\overrightarrow \beta_2 = \widehat i + 2\widehat j - 4\widehat k + \frac{4}{5}\widehat i + \frac{3}{5}\widehat j + \widehat k$ $\overrightarrow \beta_2 = \left(1 + \frac{4}{5}\right)\widehat i + \left(2 + \frac{3}{5}\right)\widehat j + (-4 + 1)\widehat k$ $\overrightarrow \beta_2 = \left(\frac{5+4}{5}\right)\widehat i + \left(\frac{10+3}{5}\right)\widehat j - 3\widehat k$ $\overrightarrow \beta_2 = \frac{9}{5}\widehat i + \frac{13}{5}\widehat j - 3\widehat k$

Step 4: Calculate the Final Required Value We need to find the value of $5{\overrightarrow \beta _2}\,.\left( {\widehat i + \widehat j + \widehat k} \right)$. First, let's compute the dot product $\overrightarrow \beta_2 \cdot (\widehat i + \widehat j + \widehat k)$: $\overrightarrow \beta_2 \cdot (\widehat i + \widehat j + \widehat k) = \left(\frac{9}{5}\widehat i + \frac{13}{5}\widehat j - 3\widehat k\right) \cdot (\widehat i + \widehat j + \widehat k)$ $= \left(\frac{9}{5}\right)(1) + \left(\frac{13}{5}\right)(1) + (-3)(1)$ $= \frac{9}{5} + \frac{13}{5} - 3$ $= \frac{22}{5} - 3$ Find a common denominator to perform the subtraction: $= \frac{22}{5} - \frac{15}{5}$ $= \frac{7}{5}$ Finally, multiply this result by 5: $5 \times \left(\frac{7}{5}\right) = 7$

Common Mistakes & Tips

• Arithmetic Errors: Be extremely careful with fraction arithmetic and signs during vector addition, subtraction, and scalar multiplication.
• Projection Formula: Ensure you are using the correct formula for vector projection, not scalar projection.
• Verification: As a good practice, you can verify that $\overrightarrow \beta_2$ is indeed perpendicular to $\overrightarrow \alpha$ by checking if their dot product is zero. This was done in the scratchpad and confirmed the correctness of $\overrightarrow \beta_2$.

Summary The problem involves decomposing a vector $\overrightarrow \beta$ into components parallel and perpendicular to another vector $\overrightarrow \alpha$. We calculated the parallel component $\overrightarrow \beta_1$ using the vector projection formula. Then, we found the perpendicular component $\overrightarrow \beta_2$ by subtracting $\overrightarrow \beta_1$ from $\overrightarrow \beta$. Finally, we computed the dot product of $\overrightarrow \beta_2$ with the vector $(\widehat i + \widehat j + \widehat k)$ and multiplied the result by 5 to obtain the final answer.

The final answer is $\boxed{7}$.