Certainly! Let's solve the problem step-by-step to find the angle between the vectors [tex]\( v = 3i - j \)[/tex] and [tex]\( w = 4i + 7j \)[/tex].
### Step 1: Represent the vectors
First, express the vectors in component form:
[tex]\[ v = \begin{pmatrix} 3 \\ -1 \end{pmatrix}, \quad w = \begin{pmatrix} 4 \\ 7 \end{pmatrix} \][/tex]
### Step 2: Calculate the dot product
The dot product [tex]\( v \cdot w \)[/tex] of two vectors is given by:
[tex]\[ v \cdot w = v_x \cdot w_x + v_y \cdot w_y \][/tex]
For our vectors:
[tex]\[ v \cdot w = (3)(4) + (-1)(7) = 12 - 7 = 5 \][/tex]
### Step 3: Calculate the magnitudes of the vectors
The magnitude (or length) of a vector [tex]\( v = \begin{pmatrix} v_x \\ v_y \end{pmatrix} \)[/tex] is given by:
[tex]\[ \|v\| = \sqrt{v_x^2 + v_y^2} \][/tex]
For vector [tex]\( v \)[/tex]:
[tex]\[ \|v\| = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.162 \][/tex]
For vector [tex]\( w \)[/tex]:
[tex]\[ \|w\| = \sqrt{4^2 + 7^2} = \sqrt{16 + 49} = \sqrt{65} \approx 8.062 \][/tex]
### Step 4: Calculate the cosine of the angle
The cosine of the angle [tex]\( \theta \)[/tex] between two vectors is given by:
[tex]\[ \cos \theta = \frac{v \cdot w}{\|v\| \|w\|} \][/tex]
Substitute the dot product and magnitudes:
[tex]\[ \cos \theta = \frac{5}{3.162 \cdot 8.062} \approx \frac{5}{25.507} \approx 0.196 \][/tex]
### Step 5: Calculate the angle in radians
The angle [tex]\( \theta \)[/tex] can be found by taking the inverse cosine (arccos) of the cosine value:
[tex]\[ \theta = \arccos(0.196) \][/tex]
This gives:
[tex]\[ \theta \approx 1.373 \text{ radians} \][/tex]
### Step 6: Convert the angle to degrees
To convert radians to degrees, use the conversion factor [tex]\( 180^\circ/\pi \)[/tex]:
[tex]\[ \theta_{\text{degrees}} = \theta_{\text{radians}} \cdot \frac{180^\circ}{\pi} \][/tex]
[tex]\[ \theta_{\text{degrees}} \approx 1.373 \cdot \frac{180^\circ}{3.14159} \approx 78.690^\circ \][/tex]
Thus, the angle between the vectors [tex]\( v \)[/tex] and [tex]\( w \)[/tex] is approximately [tex]\( 78.69^\circ \)[/tex].
