Find a + b, 9a + 7b, |a|, and |a - b|. a = 9i - 8j + 7k, b = 7i - 9k

Solution:

a and b are the two given vectors. We will use the values of a and b to find a + b, 9a + 7b, |a|, and |a - b|.

Let us solve these.

1) a + b

Here we are adding two vectors.

⇒ 9i - 8j + 7k + (7i - 9k)

Add the like terms,

16i - 8j - 2k

2) 9a + 7b

⇒ 9(9i - 8j + 7k) + 7(7i - 9k)

⇒ 81i - 72j + 63k + 49i - 63k

Add the like terms,

⇒ 130i - 72j

3) |a|

|a| = a if a ≥ 0 and |a| = - a if a < 0.

Since, |a| = √a² which represents the unique positive number. Using this magnitude formula

⇒ |a| = √ (x² + y² + z²) where x = 9i, y = 8j and z = 7k

⇒ |a| = √9² + 8² + 7²

⇒ |a| = √81 + 64 + 49

⇒ |a| = √194

⇒ |a| = 13.92

4) |a - b|.

Here we are subtracting two vectors.

|a - b| = (a - b) if a - b ≥ 0 and |a| = - (a- b) if a - b < 0.

Since, |a- b| = √(a - b)² which represents the unique positive number,

a - b = 9i - 8j + 7k - (7i - 9k)

a - b = 9i - 8j + 7k - 7i + 9k

a - b = 2i - 8j + 16k

⇒ |a - b| = √ (x² + y² + z²) where xi = 2, yi = 8 and zi = 16k

⇒ |a - b| = √2² + 8² + 16²

⇒ |a - b| = √4 + 64 + 256

⇒ |a - b| = √324

⇒ |a - b| = 18

---

Find a + b, 9a + 7b, |a|, and |a - b|. a = 9i - 8j + 7k, b = 7i - 9k

Summary:

The values of a + b = 16i - 8j - 2k, 9a + 7b = 130i - 72j, |a| = 13.92, and |a - b| = 18,if a = 9i - 8j + 7k, b = 7i - 9k.