`vec(A)+vec(B)=2hat(i)` and `vec(A)- vec(B)=4hat(j)` then angle between `vec(A)` and `vec(B)` is ?

Explanation

To solve the problem, we need to find the angle between the vectors \(\vec{A}\) and \(\vec{B}\) given the equations:

1. \(\vec{A} + \vec{B} = 2\hat{i}\)
2. \(\vec{A} - \vec{B} = 4\hat{j}\)

### Step 1: Add the two equations

We start by adding the two equations to eliminate \(\vec{B}\):

\[ (\vec{A} + \vec{B}) + (\vec{A} - \vec{B}) = 2\hat{i} + 4\hat{j} \]

This simplifies to:

\[ 2\vec{A} = 2\hat{i} + 4\hat{j} \]

### Step 2: Solve for \(\vec{A}\)

Now, divide both sides by 2:

\[ \vec{A} = \hat{i} + 2\hat{j} \]

### Step 3: Substitute \(\vec{A}\) back to find \(\vec{B}\)

Now, substitute \(\vec{A}\) back into one of the original equations to find \(\vec{B}\). We can use the first equation:

\[ \hat{i} + 2\hat{j} + \vec{B} = 2\hat{i} \]

Rearranging gives:

\[ \vec{B} = 2\hat{i} - (\hat{i} + 2\hat{j}) = \hat{i} - 2\hat{j} \]

### Step 4: Calculate the dot product \(\vec{A} \cdot \vec{B}\)

Now we have:

\[ \vec{A} = \hat{i} + 2\hat{j} \]
\[ \vec{B} = \hat{i} - 2\hat{j} \]

We calculate the dot product \(\vec{A} \cdot \vec{B}\):

\[ \vec{A} \cdot \vec{B} = (1)(1) + (2)(-2) = 1 - 4 = -3 \]

### Step 5: Calculate the magnitudes of \(\vec{A}\) and \(\vec{B}\)

Now we find the magnitudes:

\[ |\vec{A}| = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

\[ |\vec{B}| = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Step 6: Use the cosine formula to find the angle

Using the formula for the cosine of the angle \(\theta\) between two vectors:

\[ \cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} \]

Substituting the values we found:

\[ \cos \theta = \frac{-3}{\sqrt{5} \cdot \sqrt{5}} = \frac{-3}{5} \]

### Step 7: Calculate the angle \(\theta\)

Now we find \(\theta\):

\[ \theta = \cos^{-1}\left(-\frac{3}{5}\right) \]

This gives us the angle between \(\vec{A}\) and \(\vec{B}\). The approximate value of \(\theta\) is:

\[ \theta \approx 127^\circ \]

### Final Answer

The angle between \(\vec{A}\) and \(\vec{B}\) is approximately \(127^\circ\).

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