If `vecB` is added to `vecC = 3. 0 hati + 4. 0 hatj,` the result is a vector in the positive direction of the y axis, with a magnitude equal to that of `vecC.` What is the magnitude of `vecB` ?

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We know that vector `vecC` is given as:
\[ \vec{C} = 3.0 \hat{i} + 4.0 \hat{j} \]
When vector `vecB` is added to `vecC`, the resultant vector points in the positive direction of the y-axis and has a magnitude equal to that of `vecC`.

### Step 2: Write the expression for the resultant vector
Let vector `vecB` be represented as:
\[ \vec{B} = a \hat{i} + b \hat{j} \]
The resultant vector `\vec{R}` when `vecB` is added to `vecC` is:
\[ \vec{R} = \vec{B} + \vec{C} = (a + 3) \hat{i} + (b + 4) \hat{j} \]

### Step 3: Analyze the direction of the resultant vector
Since the resultant vector `\vec{R}` is in the positive direction of the y-axis, its x-component must be zero. Therefore, we can set up the equation:
\[ a + 3 = 0 \]
From this, we can solve for `a`:
\[ a = -3 \]

### Step 4: Determine the magnitude of the resultant vector
The magnitude of vector `vecC` is calculated as follows:
\[ |\vec{C}| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 5: Set the magnitude of the resultant vector equal to the magnitude of `vecC`
The magnitude of the resultant vector `\vec{R}` is given by:
\[ |\vec{R}| = \sqrt{(a + 3)^2 + (b + 4)^2} \]
Since we already found that \( a + 3 = 0 \), we can simplify this to:
\[ |\vec{R}| = \sqrt{(0)^2 + (b + 4)^2} = |b + 4| \]
According to the problem, this magnitude must equal the magnitude of `vecC`, which is 5:
\[ |b + 4| = 5 \]

### Step 6: Solve for `b`
This absolute value equation gives us two cases:
1. \( b + 4 = 5 \)
2. \( b + 4 = -5 \)

**Case 1:**
\[ b + 4 = 5 \implies b = 1 \]

**Case 2:**
\[ b + 4 = -5 \implies b = -9 \]

### Step 7: Determine the magnitude of `vecB`
Now we have two possible values for `b`: \( b = 1 \) or \( b = -9 \). We will find the magnitudes of `vecB` for both cases.

1. If \( b = 1 \):
\[ \vec{B} = -3 \hat{i} + 1 \hat{j} \]
The magnitude of `vecB` is:
\[ |\vec{B}| = \sqrt{(-3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10} \]

2. If \( b = -9 \):
\[ \vec{B} = -3 \hat{i} - 9 \hat{j} \]
The magnitude of `vecB` is:
\[ |\vec{B}| = \sqrt{(-3)^2 + (-9)^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10} \]

### Conclusion
The magnitude of `vecB` can either be \( \sqrt{10} \) or \( 3\sqrt{10} \). However, since the problem does not specify which vector direction is preferred, we will consider the magnitude of `vecB` as \( \sqrt{10} \) for the case where \( b = 1 \).

### Final Answer
The magnitude of `vecB` is:
\[ |\vec{B}| = \sqrt{10} \approx 3.16 \]