Consider two vectors `vecF_(1)=2hati+5hatk` and `vecF_(2)=3hatj+4hatk`. The magnitude to thhe scalar product of these vectors is

Options

A. 20
B. 23
C. `5sqrt(33)`
D. 26

Answer

To find the magnitude of the scalar product (dot product) of the two vectors \(\vec{F_1} = 2\hat{i} + 5\hat{k}\) and \(\vec{F_2} = 3\hat{j} + 4\hat{k}\), we will follow these steps:

### Step 1: Identify the components of the vectors
- The first vector \(\vec{F_1}\) has components:
- \(F_{1x} = 2\) (coefficient of \(\hat{i}\))
- \(F_{1y} = 0\) (no \(\hat{j}\) component)
- \(F_{1z} = 5\) (coefficient of \(\hat{k}\))

- The second vector \(\vec{F_2}\) has components:
- \(F_{2x} = 0\) (no \(\hat{i}\) component)
- \(F_{2y} = 3\) (coefficient of \(\hat{j}\))
- \(F_{2z} = 4\) (coefficient of \(\hat{k}\))

### Step 2: Use the formula for the scalar product
The scalar product (dot product) of two vectors \(\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}\) and \(\vec{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}\) is given by:
\[ \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z \]

### Step 3: Substitute the components into the formula
For our vectors:
\[ \vec{F_1} \cdot \vec{F_2} = (2)(0) + (0)(3) + (5)(4) \]

### Step 4: Calculate each term
- The first term: \(2 \times 0 = 0\)
- The second term: \(0 \times 3 = 0\)
- The third term: \(5 \times 4 = 20\)

### Step 5: Sum the results
Now, we sum the results of the three terms:
\[ \vec{F_1} \cdot \vec{F_2} = 0 + 0 + 20 = 20 \]

### Step 6: Determine the magnitude of the scalar product
Since the scalar product is a scalar quantity, its magnitude is simply:
\[ |\vec{F_1} \cdot \vec{F_2}| = 20 \]

### Final Answer
The magnitude of the scalar product of the vectors \(\vec{F_1}\) and \(\vec{F_2}\) is \(20\).

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