Question

Find the unit vector along the vector \( \vec{a} = 2\hat{i} + 3\hat{j} + \hat{k} \).

Hint:

Always double-check your calculation of the magnitude, as it is a common place for errors. The components of the final unit vector are the direction cosines of the original vector.

Answer 1

Step 1: Understanding the Concept:
A unit vector is a vector with a magnitude (or length) of 1. To find the unit vector in the direction of a given vector, we divide the vector by its magnitude.
Step 2: Key Formula or Approach:
The formula for the unit vector \( \hat{a} \) in the direction of vector \( \vec{a} \) is: \[ \hat{a} = \frac{\vec{a}}{|\vec{a}|} \] where \( |\vec{a}| \) is the magnitude of \( \vec{a} \).
Step 3: Detailed Explanation:
The given vector is \( \vec{a} = 2\hat{i} + 3\hat{j} + \hat{k} \).
First, we calculate the magnitude of \( \vec{a} \): \[ |\vec{a}| = \sqrt{(2)^2 + (3)^2 + (1)^2} \] \[ |\vec{a}| = \sqrt{4 + 9 + 1} = \sqrt{14} \] Now, we divide the vector \( \vec{a} \) by its magnitude \( |\vec{a}| \) to get the unit vector \( \hat{a} \): \[ \hat{a} = \frac{2\hat{i} + 3\hat{j} + \hat{k}}{\sqrt{14}} \] This can also be written as: \[ \hat{a} = \frac{2}{\sqrt{14}}\hat{i} + \frac{3}{\sqrt{14}}\hat{j} + \frac{1}{\sqrt{14}}\hat{k} \] Step 4: Final Answer:
The unit vector along \( \vec{a} \) is \( \frac{1}{\sqrt{14}}(2\hat{i} + 3\hat{j} + \hat{k}) \).