The component forms of vectors u and v are given by `u = u = (:5,3:) and v = (:2,-7:)`. Given that `2u + (-3v) + w = 0`, what is the component form of w?

Options

A. `(: - 16, 15:)`
B. `(: - 4, -27:)`
C. `(: 3, 10:)`
D. `(: 4, 27:)`

Answer

To find the component form of vector \( w \) given the equation \( 2u + (-3v) + w = 0 \), we will follow these steps:

### Step 1: Identify the component forms of vectors \( u \) and \( v \)
The component forms are given as:
- \( u = (5, 3) \)
- \( v = (2, -7) \)

### Step 2: Write the equation in vector form
We can express the equation \( 2u + (-3v) + w = 0 \) as:
\[ 2u - 3v + w = 0 \]

### Step 3: Substitute the component forms of \( u \) and \( v \)
Substituting the values of \( u \) and \( v \):
\[ 2(5, 3) - 3(2, -7) + w = 0 \]

### Step 4: Calculate \( 2u \) and \( -3v \)
Calculating \( 2u \):
\[ 2u = (2 \cdot 5, 2 \cdot 3) = (10, 6) \]
Calculating \( -3v \):
\[ -3v = (-3 \cdot 2, -3 \cdot -7) = (-6, 21) \]

### Step 5: Combine \( 2u \) and \( -3v \)
Now, we combine \( 2u \) and \( -3v \):
\[ (10, 6) + (-6, 21) = (10 - 6, 6 + 21) = (4, 27) \]

### Step 6: Substitute back into the equation
Now substitute back into the equation:
\[ (4, 27) + w = 0 \]

### Step 7: Solve for \( w \)
To find \( w \), we rearrange the equation:
\[ w = - (4, 27) = (-4, -27) \]

### Conclusion
Thus, the component form of \( w \) is:
\[ w = (-4, -27) \]

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