Final answer:
To prove x=y=z=0 when xa+yb+zc=0 with a, b, and c being non-zero mutually orthogonal vectors, we use their dot products. Since a, b, and c are orthogonal, the only scalars that satisfy the equation are zero. The dot product of the equation with a, b, and c individually confirms that x, y, and z must all be zero.
Step-by-step explanation:
To show that x=y=z=0 for the equation xa+yb+zc=0 when a, b, and c are non-zero mutually orthogonal vectors, we will use the concept of the dot product (scalar product) in vector algebra. If the vectors are orthogonal, their dot product is zero. Since the vectors are also non-zero, the only way the sum can be the null vector is if each scalar coefficient (x, y, z) is zero.
Assuming that xa+yb+zc=0, we take the dot product of both sides of the equation with each of the vectors a, b, and c. For example, taking the dot product of both sides of the equation with a, we get:
(xa+yb+zc) · a = 0 · a
x(a · a) + y(b · a) + z(c · a) = 0
Since a · a ≠ 0 (as a is not the null vector) and b · a = c · a = 0 (since a, b, c are mutually orthogonal), we have:
x(a · a) + 0 + 0 = 0
Therefore, x must be 0. Repeat this process with b and c to find that y and z must also be zero.