If ∣x∣=8, ∣y∣=13 and ∣x+y∣=15, determine ∣x-y∣. You may want to draw a diagram to help with your solution.
say x = 8 i + 0 j
y = a i + b j
x+y = (a+8) i + b j
so
a^2+b^2 = 169
(a+8)^2 + b ^2 = 225
a^2 + 16 a + 64 + 169 -a^2 = 225
etc
i don't understand
To determine the value of ∣x-y∣, we can use the triangle inequality.
The triangle inequality states that for any two real numbers a and b, the absolute value of the difference between them (∣a - b∣) is less than or equal to the sum of their absolute values (∣a∣ + ∣b∣). Mathematically, this can be represented as follows:
∣a - b∣ ≤ ∣a∣ + ∣b∣
In our case, we are given that ∣x+y∣ = 15, ∣x∣ = 8, and ∣y∣ = 13. Let's use these values to find ∣x-y∣.
Since ∣x+y∣ = 15, we can rewrite it as ∣(-y+x)∣ = 15. This means that the distance between x and -y is equal to 15.
Now, let's consider the triangle inequality with a = x, b = -y.
∣x - (-y)∣ ≤ ∣x∣ + ∣-y∣
This simplifies to:
∣x + y∣ ≤ ∣x∣ + ∣y∣
Since ∣x + y∣ = 15, and we know that ∣x∣ = 8 and ∣y∣ = 13, we have:
15 ≤ 8 + 13
15 ≤ 21
This inequality holds true.
Since the triangle inequality holds true, it means that ∣x-y∣ is less than or equal to ∣x∣ + ∣-y∣, which is the same as ∣x∣ + ∣y∣.
Therefore, ∣x-y∣ ≤ 8 + 13 = 21.
The maximum value for ∣x-y∣ is 21.
I hope this explanation helps!