Find the vector `vecX` which when added to the vector `vecY=6hati-7hatj` in a plane gives a resultant vectors of magnitude 5 units , along the negative y direction .

Answer

To find the vector \(\vec{X}\) that, when added to the vector \(\vec{Y} = 6\hat{i} - 7\hat{j}\), results in a vector of magnitude 5 units directed along the negative y-axis, we can follow these steps:

### Step 1: Define the resultant vector
The resultant vector \(\vec{R}\) is given to be of magnitude 5 units in the negative y-direction. Therefore, we can express it as:
\[ \vec{R} = 0\hat{i} - 5\hat{j} = -5\hat{j} \]

### Step 2: Write the equation for the resultant vector
The resultant vector \(\vec{R}\) can also be expressed as the sum of \(\vec{X}\) and \(\vec{Y}\):
\[ \vec{R} = \vec{X} + \vec{Y} \]
Substituting \(\vec{Y}\) into the equation gives:
\[ \vec{R} = \vec{X} + (6\hat{i} - 7\hat{j}) \]

### Step 3: Set up the equation
We can express \(\vec{X}\) in terms of its components:
\[ \vec{X} = a\hat{i} + b\hat{j} \]
Substituting this into the equation gives:
\[ -5\hat{j} = (a\hat{i} + b\hat{j}) + (6\hat{i} - 7\hat{j}) \]

### Step 4: Combine like terms
Combining the components on the right side, we have:
\[ -5\hat{j} = (a + 6)\hat{i} + (b - 7)\hat{j} \]

### Step 5: Equate the coefficients
Since the left side has no \(\hat{i}\) component, the coefficient of \(\hat{i}\) on the right must be zero:
\[ a + 6 = 0 \quad \text{(1)} \]
For the \(\hat{j}\) component, we equate:
\[ b - 7 = -5 \quad \text{(2)} \]

### Step 6: Solve the equations
From equation (1):
\[ a + 6 = 0 \implies a = -6 \]

From equation (2):
\[ b - 7 = -5 \implies b = 2 \]

### Step 7: Write the vector \(\vec{X}\)
Now that we have the values of \(a\) and \(b\), we can write the vector \(\vec{X}\):
\[ \vec{X} = -6\hat{i} + 2\hat{j} \]

### Final Answer
Thus, the vector \(\vec{X}\) is:
\[ \vec{X} = -6\hat{i} + 2\hat{j} \]

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