Question

The projection of $\vec{a} = \hat{\text{i}} - 2\hat{\text{j}} + \hat{\text{k}}$ on $\vec{b} = 2\hat{\text{i}} - \hat{\text{j}} + \hat{\text{k}}$ is

• A. $5$
• B. $\frac{5}{6}$
• C. $\frac{5}{\sqrt{6}}$
• D. $\sqrt{6}$

Hint:

Remember the ordering rule to avoid swapping your terms: the vector you are projecting ON is always the one whose magnitude goes in the denominator! Since we are projecting on $\vec{b}$, $|\vec{b}|$ must divide the dot product.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the scalar projection length of vector $\vec{a}$ onto the direction line of vector $\vec{b}$.

Step 2: Key Formula or Approach:
The mathematical scalar projection of a vector $\vec{a}$ onto a vector $\vec{b}$ is given by the formula:
$$\text{Projection} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$$

Step 3: Detailed Explanation:
Let's compute the components of our formula step-by-step:
1. Calculate the dot product $\vec{a} \cdot \vec{b}$:
$$\vec{a} \cdot \vec{b} = (1)(2) + (-2)(-1) + (1)(1)$$ $$\vec{a} \cdot \vec{b} = 2 + 2 + 1 = 5$$ 2. Calculate the magnitude of vector $\vec{b}$:
$$|\vec{b}| = \sqrt{(2)^2 + (-1)^2 + (1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6}$$ 3. Combine these two values into our projection formula:
$$\text{Projection} = \frac{5}{\sqrt{6}}$$ This matches option (C).

Step 4: Final Answer:
The projection of vector $\vec{a}$ on vector $\vec{b}$ is $\frac{5}{\sqrt{6}}$, which corresponds to option (C).