calculate r=a-b+c where a=5i+4j+2k b=-2i+2j+3k.calculate the angle between r and the z axis .find the angle between a and b?

To tackle this problem, we need to break it down into manageable parts. We will first calculate the vector \( r \) using the given vectors \( a \) and \( b \), then find the angle between \( r \) and the z-axis, and finally determine the angle between vectors \( a \) and \( b \).

Step 1: Calculate the Vector \( r \)

Given the vectors:

• \( a = 5i + 4j + 2k \)
• \( b = -2i + 2j + 3k \)

We can find \( r \) using the formula:

\( r = a - b + c \)

However, we notice that \( c \) is not defined in your question. Assuming \( c = 0 \) for simplicity, we can calculate \( r \) as follows:

First, calculate \( a - b \):

\( a - b = (5i + 4j + 2k) - (-2i + 2j + 3k) \)

\( = (5 + 2)i + (4 - 2)j + (2 - 3)k \)

\( = 7i + 2j - 1k \)

Thus, we have:

\( r = 7i + 2j - 1k \)

Step 2: Find the Angle Between \( r \) and the z-axis

The z-axis can be represented by the unit vector \( k = 0i + 0j + 1k \). To find the angle \( \theta \) between \( r \) and the z-axis, we can use the dot product formula:

\( \cos(\theta) = \frac{r \cdot k}{|r| |k|} \)

First, calculate the dot product \( r \cdot k \):

\( r \cdot k = (7i + 2j - 1k) \cdot (0i + 0j + 1k) = -1 \)

Next, find the magnitudes of \( r \) and \( k \):

\( |r| = \sqrt{7^2 + 2^2 + (-1)^2} = \sqrt{49 + 4 + 1} = \sqrt{54} \)

\( |k| = 1 \)

Now, substituting these values into the cosine formula:

\( \cos(\theta) = \frac{-1}{\sqrt{54} \cdot 1} = \frac{-1}{\sqrt{54}} \)

To find \( \theta \), take the inverse cosine:

\( \theta = \cos^{-1}\left(\frac{-1}{\sqrt{54}}\right) \)

Step 3: Calculate the Angle Between Vectors \( a \) and \( b \)

To find the angle \( \phi \) between vectors \( a \) and \( b \), we again use the dot product:

\( \cos(\phi) = \frac{a \cdot b}{|a| |b|} \)

First, calculate the dot product \( a \cdot b \):

\( a \cdot b = (5i + 4j + 2k) \cdot (-2i + 2j + 3k) \)

\( = (5 \cdot -2) + (4 \cdot 2) + (2 \cdot 3) = -10 + 8 + 6 = 4 \)

Next, find the magnitudes of \( a \) and \( b \):

\( |a| = \sqrt{5^2 + 4^2 + 2^2} = \sqrt{25 + 16 + 4} = \sqrt{45} \)

\( |b| = \sqrt{(-2)^2 + 2^2 + 3^2} = \sqrt{4 + 4 + 9} = \sqrt{17} \)

Now, substituting these values into the cosine formula:

\( \cos(\phi) = \frac{4}{\sqrt{45} \cdot \sqrt{17}} \)

Finally, to find \( \phi \), take the inverse cosine:

\( \phi = \cos^{-1}\left(\frac{4}{\sqrt{765}}\right) \)

Summary of Results

In summary, we calculated:

• \( r = 7i + 2j - 1k \)
• The angle \( \theta \) between \( r \) and the z-axis is \( \cos^{-1}\left(\frac{-1}{\sqrt{54}}\right) \)
• The angle \( \phi \) between vectors \( a \) and \( b \) is \( \cos^{-1}\left(\frac{4}{\sqrt{765}}\right) \)

These calculations provide a comprehensive understanding of the relationships between the vectors in question. If you have any further questions or need clarification on any steps, feel free to ask!