Let vector a and b be two non-null vectors such that |vector a +b| =|vector a-2b| then the values of |vector a| / |vector b| may be:
To find the possible values of the magnitude of vector a divided by the magnitude of vector b, we can start by using the given information. Let's break down the problem step by step:
1. Start with the given condition: |vector a + vector b| = |vector a - 2vector b|
2. Expand both sides using the properties of vector addition and scalar multiplication:
|vector a| + |vector b| + 2|vector a| * |vector b| * cos(theta) = |vector a| - 2|vector b|,
where theta is the angle between vectors a and b.
3. Simplify the equation by canceling out the common terms:
|vector b| + 2|vector a| * |vector b| * cos(theta) = -2|vector b|.
4. Re-arrange the equation to isolate the cos(theta) term:
2|vector a| * |vector b| * cos(theta) = -3|vector b|.
5. Divide both sides by 2|vector a| * |vector b|:
cos(theta) = (-3|vector b|) / (2|vector a| * |vector b|).
6. Simplify the equation further:
cos(theta) = -3 / (2|vector a|).
7. Since cos(theta) lies between -1 and 1, we have:
-1 ≤ cos(theta) ≤ 1.
8. Substitute the value of cos(theta) from the previous step:
-1 ≤ -3 / (2|vector a|) ≤ 1.
9. Now, let's solve for |vector a| / |vector b|, which is the ratio we're interested in. Multiply throughout by 2 and rearrange the inequality:
-2 ≤ -3 / (|vector a|) ≤ 2.
10. Multiply throughout by |vector a|, remembering that |vector a| ≠ 0:
-2|vector a| ≤ -3 ≤ 2|vector a|.
11. Divide throughout by 3:
-2/3 ≤ -1/3|vector a| ≤ 2/3.
12. Divide throughout by (-1/3):
2 ≤ |vector a| / |vector b| ≤ 6/2.
Therefore, the possible values of |vector a| / |vector b| are from 2 to 3.
Note: While the above steps provide a general solution, it's important to double-check the algebraic operations and any assumptions made in the problem statement.