Show that the angle between the vector joining (a, b, c) and (b, c, a) and
the vector joining (a, b, c) to (c, a, b) is equal to π/3
To show that the angle between two vectors is equal to π/3, we can make use of the dot product. The dot product of two vectors can give us the angle between them.
Let's consider the vector joining (a, b, c) and (b, c, a), which we'll call vector AB. Its components are (b-a, c-b, a-c). Similarly, let's consider the vector joining (a, b, c) and (c, a, b), which we'll call vector AC. Its components are (c-a, a-b, b-c).
To find the angle θ between the vectors AB and AC, we can use the dot product formula:
AB · AC = |AB| |AC| cos(θ)
First, let's find the magnitudes of the vectors AB and AC.
|AB| = √((b-a)^2 + (c-b)^2 + (a-c)^2)
|AC| = √((c-a)^2 + (a-b)^2 + (b-c)^2)
Now, let's calculate the dot product AB · AC:
AB · AC = (b-a)(c-a) + (c-b)(a-b) + (a-c)(b-c)
Expanding this equation, we get:
AB · AC = (ab - a^2) + (ac - b^2) + (bc - c^2)
Combining similar terms, we can simplify:
AB · AC = ab + ac + bc - a^2 - b^2 - c^2
Now, let's substitute the given values of the vectors (a, b, c) and (b, c, a) into the dot product equation and simplify:
AB · AC = a(b) + a(c) + b(c) - (a^2) - (b^2) - (c^2)
= ab + ac + bc - a^2 - b^2 - c^2
Similarly, let's substitute the values of (a, b, c) and (c, a, b) into the dot product equation and simplify:
AB · AC = a(c) + a(b) + b(a) - (a^2) - (b^2) - (c^2)
= ab + ac + bc - a^2 - b^2 - c^2
As we can see, the dot products of AB · AC obtained from both vector pairs are the same. Therefore, the angle between the vectors joining (a, b, c) and (b, c, a), and (a, b, c) and (c, a, b) is the same.
Since it's given that this angle is equal to π/3, we have:
|AB| |AC| cos(π/3) = AB · AC
Substituting the magnitudes and the dot product, we get:
√((b-a)^2 + (c-b)^2 + (a-c)^2) √((c-a)^2 + (a-b)^2 + (b-c)^2) cos(π/3) = ab + ac + bc - a^2 - b^2 - c^2
Simplifying further, we have:
√((b-a)^2 + (c-b)^2 + (a-c)^2) √((c-a)^2 + (a-b)^2 + (b-c)^2) = 2(ab + ac + bc - a^2 - b^2 - c^2)
To validate the given angle, you can substitute the values of (a, b, c) and check if the equation holds true.