Question

Let \[ \vec a=\hat i+2\hat j+2\hat k, \qquad \vec b=2\hat i+\hat j+2\hat k. \] If \(\theta\) is the angle between \(\vec a\) and \(\vec b\), then the value of \[ \cos\theta \] is:

• A. \(\frac{8}{9}\)
• B. \(\frac{7}{9}\)
• C. \(\frac{5}{9}\)
• D. \(\frac{4}{9}\)

Hint:

For vectors \[ (a,b,c) \quad\text{and}\quad (p,q,r), \] always remember \[ \cos\theta = \frac{ap+bq+cr} {\sqrt{a^2+b^2+c^2}\sqrt{p^2+q^2+r^2}}. \] This is the standard formula for finding the angle between two vectors.

Answer 1

The Correct Option is A

Step 1: Recall the angle formula.
For two vectors, \[ \cos\theta=\frac{\vec a\cdot\vec b}{|\vec a|\,|\vec b|}. \] We just need the dot product and the two lengths.

Step 2: Write components.
$\vec a=(1,2,2)$ and $\vec b=(2,1,2)$.

Step 3: Find the dot product.
\[ \vec a\cdot\vec b=(1)(2)+(2)(1)+(2)(2)=2+2+4=8. \]

Step 4: Find $|\vec a|$.
\[ |\vec a|=\sqrt{1^2+2^2+2^2}=\sqrt{1+4+4}=\sqrt9=3. \]

Step 5: Find $|\vec b|$.
\[ |\vec b|=\sqrt{2^2+1^2+2^2}=\sqrt{4+1+4}=\sqrt9=3. \]

Step 6: Combine.
\[ \cos\theta=\frac{8}{3\times3}=\frac{8}{9}. \] This is option 1.
\[ \boxed{\dfrac{8}{9}} \]