Calculate the dot product of \( \vec{u}=<-10,20,-13> \) and \( \vec{v}=<-5,0,-2> \) Show all steps.

To calculate the dot product of two vectors \( \vec{u} \) and \( \vec{v} \), we use the formula: \[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \] Given vectors \( \vec{u} = <-10, 20, -13> \) and \( \vec{v} = <-5, 0, -2> \), we can substitute the components into the formula: \[ \vec{u} \cdot \vec{v} = (-10)(-5) + (20)(0) + (-13)(-2) \] Now, let's calculate the dot product step by step: Calculate the value by following steps: - step0: Calculate: \(\left(-10\right)\left(-5\right)+20\times 0+\left(-13\right)\left(-2\right)\) - step1: Multiply: \(\left(-10\right)\left(-5\right)+0+\left(-13\right)\left(-2\right)\) - step2: Remove the parentheses: \(-10\left(-5\right)+0-13\left(-2\right)\) - step3: Multiply the numbers: \(50+0-13\left(-2\right)\) - step4: Multiply the numbers: \(50+0+26\) - step5: Remove 0: \(50+26\) - step6: Add the numbers: \(76\) The dot product of vectors \( \vec{u} = <-10, 20, -13> \) and \( \vec{v} = <-5, 0, -2> \) is 76.