Math Snap

STEP 1

What is this asking?
Find the values of $x$ that make the angle between two vectors obtuse.
Watch out!
Remember, an obtuse angle is between 90 and 180 degrees!
Don't forget to check the dot product condition for an obtuse angle.

STEP 2

1. Define the vectors
2. Calculate the dot product
3. Set up the inequality
4. Solve for x

STEP 3

Let's call our vectors $\vec{a}$ and $\vec{b}$.
We can write them as $\vec{a} = (1, 2)$ and $\vec{b} = (x, 3)$.
Super easy!

STEP 4

Remember, the dot product of two vectors $\vec{a} = (a_1, a_2)$ and $\vec{b} = (b_1, b_2)$ is given by $\vec{a} \cdot \vec{b} = a_1 \cdot b_1 + a_2 \cdot b_2$.

STEP 5

So, for our vectors, the dot product is $\vec{a} \cdot \vec{b} = (1 \cdot x) + (2 \cdot 3) = x + 6$.
See? Not so bad!

STEP 6

For the angle between the vectors to be obtuse, the dot product must be negative.
That is, $\vec{a} \cdot \vec{b} < 0$.

STEP 7

So, we have $x + 6 < 0$.
We're almost there!

STEP 8

To solve for $x$, we just need to subtract 6 from both sides of the inequality: $x + 6 - 6 < 0 - 6$.

STEP 9

This gives us our final inequality: $x < -6$.
Boom!

The angle between the vectors is obtuse when $x < -6$.