- Identify the components of the vectors \(\vec{u} = \langle 1, -4 \rangle\) and \(\vec{v} = \langle -2, -5 \rangle\).
- Apply the dot product formula: \(\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2\).
- Perform the arithmetic calculation: \((1 \times -2) + (-4 \times -5) = -2 + 20 = 18\).
- Select the matching option, which is \(\boxed{18}\).
Explanation
- Analysis and Formula Selection
We are given two vectors, \(\vec{u} = \langle 1, -4 \rangle\) and \(\vec{v} = \langle -2, -5 \rangle\). Our task is to calculate the dot product of these two vectors. The dot product of two vectors \(\vec{a} = \langle a_1, a_2 \rangle\) and \(\vec{b} = \langle b_1, b_2 \rangle\) is defined by the formula:
\[\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2.\]
Here, \(u_1 = 1\), \(u_2 = -4\), \(v_1 = -2\), and \(v_2 = -5\).
- Calculation and Result
Now, substitute the components of vectors \(\vec{u}\) and \(\vec{v}\) into the dot product formula:
\[\vec{u} \cdot \vec{v} = (1) \times (-2) + (-4) \times (-5).\]
Multiplying the terms:
\[1 \times (-2) = -2,\]
\[(-4) \times (-5) = 20.\]
Adding these results:
\[\vec{u} \cdot \vec{v} = -2 + 20 = 18.\]
The calculated dot product is 18, which corresponds to the fourth option provided.
Examples
The concept of the dot product is fundamental in physics and engineering. For example, if a force vector \(\vec{F}\) acts on an object, and the object undergoes a displacement represented by a vector \(\vec{d}\), the work done by the force is given by the dot product \(\vec{W} = \vec{F} \cdot \vec{d}\). By calculating this, you can determine exactly how much energy is transferred in a mechanical system, which is crucial for designing engines or analyzing physical motion.