Question

for ( u=langle 4,-1
angle ) and ( v=langle-2,4
angle ), find ( u cdot v ).
( u cdot v=square )

Explanation:

Step1: Recall the dot product formula for vectors

For two vectors \(\mathbf{u}=\langle u_1, u_2
angle\) and \(\mathbf{v}=\langle v_1, v_2
angle\), the dot product \(\mathbf{u}\cdot\mathbf{v}\) is given by \(u_1v_1 + u_2v_2\).

Step2: Identify the components of the vectors

Here, \(u_1 = 4\), \(u_2=-1\), \(v_1 = -2\), and \(v_2 = 4\).

Step3: Apply the dot product formula

Substitute the values into the formula: \(\mathbf{u}\cdot\mathbf{v}=(4)(-2)+(-1)(4)\).

Step4: Perform the arithmetic operations

First, calculate \(4\times(-2)= -8\) and \((-1)\times4 = -4\). Then, add the results: \(-8+(-4)=-8 - 4=-12\).

Answer:

\(-12\)