Question

what is the magnitude of \\(\vec{v}\\)?\\(\vec{v}\\) points to (8, -2)\\(\vert \vec{v} \vert = ?\\)round your answer to the nearest hundredth.

Explanation:

Step1: Recall the magnitude formula for a vector \(\vec{v}=(x,y)\)

The magnitude of a vector \(\vec{v}\) with components \(x\) and \(y\) is given by \(|\vec{v}|=\sqrt{x^{2}+y^{2}}\). Here, the vector \(\vec{v}\) has components \(x = 8\) and \(y=-2\).

Step2: Substitute the values into the formula

Substitute \(x = 8\) and \(y=-2\) into the formula: \(|\vec{v}|=\sqrt{8^{2}+(-2)^{2}}\).

Step3: Calculate the squares

Calculate \(8^{2}=64\) and \((-2)^{2}=4\). Then the expression becomes \(|\vec{v}|=\sqrt{64 + 4}\).

Step4: Add the numbers inside the square root

Add \(64\) and \(4\): \(64+4 = 68\). So \(|\vec{v}|=\sqrt{68}\).

Step5: Calculate the square root and round

Calculate \(\sqrt{68}\approx8.2462\). Rounding to the nearest hundredth (two decimal places), we look at the third decimal place which is \(6\). Since \(6\gt5\), we round up the second decimal place. So \(\sqrt{68}\approx8.25\).

Answer:

\(8.25\)