An object is acted upon by the forces \( F_{1}=\langle 12,5,3\rangle \) and \( F_{2}=\langle 0,4,8\rangle \). Find the force \( F_{3} \) that must act on the object so that the sum of the forces is zero. \( F_{3}=\langle\square, \square \)

To find the force \( F_{3} \) that must act on the object so that the sum of the forces is zero, we need to set up an equation where the sum of the forces \( F_{1} \), \( F_{2} \), and \( F_{3} \) is equal to zero. Given: \( F_{1} = \langle 12, 5, 3 \rangle \) \( F_{2} = \langle 0, 4, 8 \rangle \) We want to find \( F_{3} = \langle x, y, z \rangle \) such that: \( F_{1} + F_{2} + F_{3} = 0 \) Let's set up the equation for each component: 1. For the x-component: \( 12 + 0 + x = 0 \) 2. For the y-component: \( 5 + 4 + y = 0 \) 3. For the z-component: \( 3 + 8 + z = 0 \) Now, we can solve these equations to find the values of \( x \), \( y \), and \( z \) for \( F_{3} \). Solve the equation by following steps: - step0: Solve for \(x\): \(12+0+x=0\) - step1: Remove 0: \(12+x=0\) - step2: Move the constant to the right side: \(x=0-12\) - step3: Remove 0: \(x=-12\) Solve the equation \( 5 + 4 + y = 0 \). Solve the equation by following steps: - step0: Solve for \(y\): \(5+4+y=0\) - step1: Add the numbers: \(9+y=0\) - step2: Move the constant to the right side: \(y=0-9\) - step3: Remove 0: \(y=-9\) Solve the equation \( 3 + 8 + z = 0 \). Solve the equation by following steps: - step0: Solve for \(z\): \(3+8+z=0\) - step1: Add the numbers: \(11+z=0\) - step2: Move the constant to the right side: \(z=0-11\) - step3: Remove 0: \(z=-11\)