Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest tenth of a degree.) $\langle 2, 1, 4 \rangle$.

Neetesh Kumar | December 18, 2024

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This is the solution to Math 1C
Assignment: 12.3 Question Number 13
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Step-by-step solution:

The direction cosines of a vector $\mathbf{v} = \langle a, b, c \rangle$ are:

1. $\cos\alpha = \frac{a}{|\mathbf{v}|}$ (angle with the $x$-axis),
2. $\cos\beta = \frac{b}{|\mathbf{v}|}$ (angle with the $y$-axis),
3. $\cos\gamma = \frac{c}{|\mathbf{v}|}$ (angle with the $z$-axis).

The direction angles are found using:
$\alpha = \cos^{-1}(\cos\alpha), \quad \beta = \cos^{-1}(\cos\beta), \quad \gamma = \cos^{-1}(\cos\gamma).$

Step 1: Find the magnitude of the vector $\mathbf{v}$:

The magnitude of $\mathbf{v} = \langle 2, 1, 4 \rangle$ is:
$|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}.$

Substitute $a = 2$, $b = 1$, and $c = 4$:
$|\mathbf{v}| = \sqrt{2^2 + 1^2 + 4^2}.$

Simplify:
$|\mathbf{v}| = \sqrt{4 + 1 + 16} = \sqrt{21}.$

Step 2: Compute the direction cosines:

1. For $\cos\alpha$ (angle with the $x$-axis):
   $\cos\alpha = \frac{a}{|\mathbf{v}|} = \frac{2}{\sqrt{21}}.$

2. For $\cos\beta$ (angle with the $y$-axis):
   $\cos\beta = \frac{b}{|\mathbf{v}|} = \frac{1}{\sqrt{21}}.$

3. For $\cos\gamma$ (angle with the $z$-axis):
   $\cos\gamma = \frac{c}{|\mathbf{v}|} = \frac{4}{\sqrt{21}}.$

Step 3: Find the direction angles:

To find $\alpha$, $\beta$, and $\gamma$, take the inverse cosine (arccos) of the direction cosines.

1. For $\alpha$:
   $\alpha = \cos^{-1}\left(\frac{2}{\sqrt{21}}\right).$

2. For $\beta$:
   $\beta = \cos^{-1}\left(\frac{1}{\sqrt{21}}\right).$

3. For $\gamma$:
   $\gamma = \cos^{-1}\left(\frac{4}{\sqrt{21}}\right).$

Using a calculator to approximate:

1. $\frac{2}{\sqrt{21}} \approx 0.436$:
   $\alpha = \cos^{-1}(0.436) \approx 64.1^\circ$.

2. $\frac{1}{\sqrt{21}} \approx 0.218$:
   $\beta = \cos^{-1}(0.218) \approx 77.4^\circ$.

3. $\frac{4}{\sqrt{21}} \approx 0.872$:
   $\gamma = \cos^{-1}(0.872) \approx 29.2^\circ$.

Final Answer:

1. Direction cosines:

• $\cos\alpha = \boxed{\frac{2}{\sqrt{21}} \approx 0.436}$

  
  

• $\cos\beta = \boxed{\frac{1}{\sqrt{21}} \approx 0.218}$

  
  

• $\cos\gamma = \boxed{\frac{4}{\sqrt{21}} \approx 0.872}$

2. Direction angles:

• $\alpha \approx \boxed{64.2^\circ}$

• $\beta \approx \boxed{77.4^\circ}$

• $\gamma \approx \boxed{29.2^\circ}$

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