The magnitude of vector a (|a|) is determined as 1 (Option C).
How to calculate the magnitude of vector a?
The magnitude of vector a is calculated by applying the formula for dot product of two vectors.
a · b = |a| · |b| · cos(θ)
The given expression.
|a + b + c| =√6
|a + b + c|² = (√6)²
(a + b + c) · (a + b + c) = 6
The expression is simplified as follows;
|a|² + |b|² + |c|² + 2(a·b + b·c + c·a) = 6
a.b = |a| · |b| cosθ
θ = π/3
θ = 180/3
θ = 60
Also, |a| = |b| = |c| = y
|a|² + |b|² + |c|² + 2(a·b + b·c + c·a) = 6
|a|² + |b|² + |c|² + 2(|a|·|b| cosθ + |b|·|c| cosθ + |c|·|a| cosθ ) = 6
y² + y² + y² + 2(y² (1/2) + y²(1/2) + y²(1/2)) = 6
y² + y² + y² + (y² + y² + y²) = 6
6y² = 6
y² = 6/6
y² = 1
y = ±√1
y = ±1
|a| = 1