Question

Question was provided via image upload.

Explanation:

Identify the vectors and options

We are asked to solve Question 30 from the image.
The problem states:
"If \(\vec{a} = (4, 3, 2)\) and if \(\vec{b} = (1, 2, -3)\) are two vectors in space, then which one of the following is not correct?"

Let's write down the given vectors:

$$ \vec{a} = (4, 3, 2) $$

$$ \vec{b} = (1, 2, -3) $$

The options are:
A. \(\vec{a} + \vec{b} = (5, 5, -1)\)
B. \(\vec{a} + 2\vec{b} = (6, 7, -4)\)
C. \(\vec{b} - \vec{a} = (-3, -1, -5)\)
D. \(\vec{a} + 2\vec{b} = (2, 1, 8)\)

Verify Option A

Let's calculate the sum of the two vectors:

$$ \vec{a} + \vec{b} = (4 + 1, 3 + 2, 2 + (-3)) = (5, 5, -1) $$

This matches Option A, so Option A is correct.

Verify Option B and Option D

Let's calculate \(\vec{a} + 2\vec{b}\):
First, find \(2\vec{b}\):

$$ 2\vec{b} = 2 \cdot (1, 2, -3) = (2, 4, -6) $$

Now, add \(\vec{a}\):

$$ \vec{a} + 2\vec{b} = (4, 3, 2) + (2, 4, -6) = (4 + 2, 3 + 4, 2 - 6) = (6, 7, -4) $$

This matches Option B, so Option B is correct.
Consequently, Option D, which claims \(\vec{a} + 2\vec{b} = (2, 1, 8)\), is incorrect.

Verify Option C

Let's calculate the difference \(\vec{b} - \vec{a}\):

$$ \vec{b} - \vec{a} = (1 - 4, 2 - 3, -3 - 2) = (-3, -1, -5) $$

This matches Option C, so Option C is correct.

Identify the incorrect statement

Since Option D states that \(\vec{a} + 2\vec{b} = (2, 1, 8)\), which contradicts our calculation of \((6, 7, -4)\), Option D is the incorrect statement.

Answer:

• (A) \(\vec{a} + \vec{b} = (5, 5, -1)\)
• (B) \(\vec{a} + 2\vec{b} = (6, 7, -4)\)
• (C) \(\vec{b} - \vec{a} = (-3, -1, -5)\)
• (D) \(\vec{a} + 2\vec{b} = (2, 1, 8)\) (Correct answer)