Understanding the Angle Between Vectors a and b When a b ab
The problem at hand involves understanding the relationship between vectors a and b given that ab is equal to both a and b. To solve this, let's delve into the properties of vectors and geometric interpretations.
Vector Dot Product and Its Application
The dot product of two vectors can be used to find the cosine of the angle between them. For vectors a and b, the formula for their dot product is given by:
( ab^2 a^2 b^2 2abcostheta )
where theta is the angle between vectors a and b. Given that (a b ab), we can substitute these values into the formula.
Mathematical Derivation
Substituting ab, a, and b into the equation:
( ab^2 a^2 b^2 2abcostheta )
Since (a b ab 1), we simplify the equation as follows:
[ 1 1 1 2 cdot 1 cdot 1 cdot cos theta ]
[ 1 2 2 cos theta ]
[ -1 2 cos theta ]
[ cos theta -frac{1}{2} ]
Therefore, [ theta arccosleft(-frac{1}{2}right) 120; degrees ]
Geometric Interpretation
From a geometric perspective, if vectors a and b have equal magnitudes and their dot product is equal to their magnitudes, such as ( vec{a} vec{b} vec{ab} ), then these vectors form the sides of an equilateral triangle. In an equilateral triangle, all angles are equal to 60 degrees, but the angle between the vectors can be described in the context of symmetry. If a third vector c of the same magnitude is 120 degrees to both a and b, then the resultant vector would be symmetric about 360 degrees with a zero resultant vector.
The vector ( vec{a} vec{b} ) is the resultant vector that connects the starting point of vector ( vec{a} ) to the ending point of vector ( vec{b} ). Drawing these vectors, one can see that they form an equilateral triangle. Therefore, the angle between them is 120 degrees.
Vector Representation and Triangle Formation
The vector ( vec{a} - vec{b} ) represents the vector that connects the starting point of vector ( vec{a} ) with the ending point of vector ( vec{b} ). By drawing these vectors, they form a triangle. Given that the lengths of the vectors are equal, it implies an equilateral triangle. The angle between the vectors a and b can be derived as follows:
( (vec{a} vec{b})^2 vec{a}^2 2 vec{a} cdot vec{b} vec{b}^2 )
Since ( vec{a} vec{b} vec{ab} ), let's simplify:
[ (vec{a} vec{b})^2 vec{a}^2 2a^2 a^2 3a^2 ]
Similarly:
[ 3a^2 a^2 2a^2 - a^2 a^2 ]
Therefore:
[ 2a^2 cos theta - a^2 0 ]
[ 2cos theta 1 ]
[ cos theta -frac{1}{2} ]
[ theta 120 ; degrees ]
Summary
In conclusion, when vectors a and b have equal magnitudes and their dot product is equal to their magnitudes, the angle between them is 120 degrees. This geometric and algebraic interpretation confirms the angle must be 120 degrees for the given conditions.
References:
MathIsFun - Vectors Khan Academy - Dot and Cross Products Explained