a, b, c are 3 vectors such that |a|=2|b|=4|c| (|a|=modulus of a vector) and 2a-3b+6c=0 (a is a vector) then find cos(x) between a and b.
To find the cosine of the angle between vectors a and b, we need to use the dot product formula. Let's break down the process step by step:
Step 1: Given vector equations
We have the following information:
|a| = 2|b| = 4|c|
2a - 3b + 6c = 0
Step 2: Expressing vectors a, b, and c in terms of one another
Since |a| = 2|b|, we can express vector a as a = 2b.
Similarly, since |b| = (1/2)|c|, we can express vector b as b = (1/2)c.
Now, we need to rewrite the equation using these expressions:
2a - 3b + 6c = 0
Substituting the expressions for a and b:
2(2b) - 3((1/2)c) + 6c = 0
4b - (3/2)c + 6c = 0
4b + (9/2)c = 0
Step 3: Finding the dot product of a and b
The dot product of two vectors a and b is defined as a · b = |a| |b| cos(x), where x is the angle between them.
Let's calculate the dot product of vectors a and b:
a · b = (2b) · b
= 2b · b
= 2|b|^2
= 2(1/2)|c|^2 (substituting |b| = (1/2)|c|)
= |c|^2
We can conclude that the dot product of a and b is equal to the square of the magnitude of vector c.
Step 4: Finding cos(x)
By substituting the dot product value back into the dot product formula, we can solve for cos(x):
a · b = |a| |b| cos(x)
|c|^2 = |a| |b| cos(x)
|c|^2 = (2|b|)(|b|) cos(x) (substituting |a| = 2|b|)
|c|^2 = 2|b|^2 cos(x)
|c|^2 = 2(cos(x) |c|)^2
1/2 = cos(x)^2
cos(x) = ±√(1/2)
Therefore, the cosine of the angle between vectors a and b is ±√(1/2).
Note: The ± sign indicates that the angle could be positive or negative depending on the direction of the vectors a and b.