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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:
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