The resultant of two vectors a and b is perpendicular to the vector a and its magnitude is equal to the half magnitude of vector b the find angle between a and b
To find the angle between vectors a and b, we can use the dot product formula.
The dot product of two vectors a and b is defined as:
a · b = |a| * |b| * cos(theta)
where |a| and |b| represent the magnitudes of vectors a and b, and theta is the angle between them.
Given that the resultant of vectors a and b is perpendicular to vector a, we know that the dot product of the resultant and vector a will be zero. This condition can be formulated as:
(a + b) · a = 0
Expanding this equation, we get:
a · a + b · a = 0
Since a · a represents the magnitude of vector a squared, and b · a represents the dot product of vectors a and b, we can rewrite the equation as:
|a|^2 + b · a = 0
Now we have an equation relating the magnitudes of a and b and their dot product. Let's apply the other given information:
It is stated that the magnitude of the resultant vector is equal to half the magnitude of vector b. Mathematically, this can be expressed as:
|a + b| = 0.5 * |b|
Squaring both sides of the equation gives:
|a + b|^2 = (0.5 * |b|)^2
(a + b) · (a + b) = (0.5 * |b|)^2
Expanding and simplifying the equation further:
a · a + 2 * a · b + b · b = 0.25 * b^2
Since a · a is the magnitude of vector a squared, and b · b is the magnitude of vector b squared, we can rewrite the equation as:
|a|^2 + 2 * a · b + |b|^2 = 0.25 * |b|^2
|a|^2 + 2 * a · b + |b|^2 = 0.25 * |b|^2
Substituting the dot product equation |a|^2 + b · a = 0 into the equation, we have:
0 + 2 * a · b + |b|^2 = 0.25 * |b|^2
Simplifying, we get:
2 * a · b = -0.75 * |b|^2
a · b = -0.375 * |b|^2
Now we have another equation relating the dot product a · b to the magnitude of vector b squared.
Using the fact that the dot product a · b can also be expressed as |a| * |b| * cos(theta), we can rewrite the equation as:
|a| * |b| * cos(theta) = -0.375 * |b|^2
Dividing both sides by |b|, we get:
|a| * cos(theta) = -0.375 * |b|
Now we can solve for the angle theta:
cos(theta) = (-0.375 * |b|) / |a|
Taking the inverse cosine of both sides, we can find the angle theta:
theta = arccos((-0.375 * |b|) / |a|)
So, to find the angle between vectors a and b, substitute the magnitudes of vectors a and b into the formula:
theta = arccos((-0.375 * |b|) / |a|)