To find the angle between two vectors $\vec{v}$ and $\vec{w}$, we can use the following formula involving the dot product:

$\cos \theta = \frac{\vec{v} \cdot \vec{w}}{\|\vec{v}\| \|\vec{w}\|}$

where:

• $\vec{v} = \begin{pmatrix} -4 \\ 3 \end{pmatrix}$
• $\vec{w} = \begin{pmatrix} 1 \\ -4 \end{pmatrix}$

Step 1: Calculate the Dot Product

The dot product $\vec{v} \cdot \vec{w}$ is given by:

$\vec{v} \cdot \vec{w} = (-4) \cdot 1 + 3 \cdot (-4) = -4 - 12 = -16$

Step 2: Calculate the Magnitudes of Each Vector

The magnitude $\|\vec{v}\|$ is:

$\|\vec{v}\| = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

The magnitude $\|\vec{w}\|$ is:

$\|\vec{w}\| = \sqrt{1^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17}$

Step 3: Substitute Values into the Formula

Now, we substitute the values into the cosine formula:

$\cos \theta = \frac{-16}{5 \cdot \sqrt{17}}$

Step 4: Calculate the Angle

To find $\theta$, take the arccosine of the result:

$\theta = \cos^{-1} \left( \frac{-16}{5 \sqrt{17}} \right)$

Let’s compute this value for an approximate answer in degrees.

Using a calculator for this:

1. Calculate the cosine value and then the arccos to get the angle.
2. This results in approximately $\theta \approx 114^\circ$ (rounded to the nearest integer).

Answer

The angle between the vectors is approximately $114^\circ$.

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How do you find the angle between vectors in 3D space?
2. What is the significance of a zero dot product between two vectors?
3. How do you calculate the projection of one vector onto another?
4. How do vectors relate to direction cosines?
5. Can two vectors be perpendicular if they have unequal magnitudes?

Tip: Remember, if the dot product is negative, the angle between the vectors is obtuse.