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Find a b, 6a 9b, |a|, and |a − b|. (simplify your vectors completely. ) a = −9, 12 , b = 6, 4

Sagot :

The values of a + b, 6a + 9b, |a|, and |a − b| are −3i + 16j, 0i + 108j, 15 and 17 respectively. This can be obtained by using vector addition, vector subtraction and formula to find magnitude of a vector.

Find the values of a + b, 6a + 9b, |a|, and |a − b|:

Given that,

a = <−9, 12> , b = <6, 4>

These vectors can be rewritten as,

a = <−9, 12> = −9i + 12j

b = <6, 4> = 6i + 4j

  • To find a + b,we add both vectors a and b together,

a + b = −9i + 12j + 6i + 4j

a + b = −9i + 6i + 12j + 4j

a + b = (−9 + 6)i + (12 + 4)j

a + b = −3i + 16j

  • To find 6a + 9b, we first find 6a and 9b then add them both together,

6a = 6 (−9i + 12j )

6a = −54i + 72j

9b = 9(6i + 4j)

9b = 54i + 36j

Now add 6a and 9b together,

6a + 9b = −54i + 72j  + 54i + 36j

6a + 9b = −54i + 54i + 72j + 36j

6a + 9b = 0i + 108j

  • To find |a|, use the formula to find the magnitude of a vector,

If a = a₁i + a₂j, |a| = √a₁² + a₂²

Here, a = −9i + 12j

|a| = √(−9)² + (12)²

|a| = √81 + 144 = √225

|a| = 15

  • To find |a − b|, first subtract b from a and find the magnitude of the resultant,

a - b = −9i + 12j - (6i + 4j)

a - b = −9i - 6i + 12j - 4j

a - b = −15i + 8j

Now use the formula to find the magnitude of a vector,

|a − b| = √(-15)² + (8)²

|a − b| = √225 + 64 = √289

|a − b| = 17

Hence the values of a + b, 6a + 9b, |a|, and |a − b| are −3i + 16j, 0i + 108j, 15 and 17 respectively.

       

Learn more about magnitude of a vector here:

brainly.com/question/27870005

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