Given A = 2i - 3j + 4k and B = i + 2j - k , what is the value of the dot product A. B?

এটি একটি গণিত বহু নির্বাচনী প্রশ্ন যা বিভিন্ন শ্রেণীর শিক্ষার্থীদের জন্য।

ব্যাখ্যা

<h1>Understanding the Dot Product of Vectors</h1>

Given vectors A = 2i - 3j + 4k and B = i + 2j - k, we need to find the value of their dot product A ⋅ B.

• -8

• 0

• 1

• 5

The dot product of two vectors <code>A</code> and <code>B</code> is defined as:

$ A \cdot B = A_x \cdot B_x + A_y \cdot B_y + A_z \cdot B_z $

Where:

• $A_x, A_y, A_z$ are the components of vector <code>A</code>

• $B_x, B_y, B_z$ are the components of vector <code>B</code>

For the given vectors, we can identify the components as follows:

Vector A: <br>$ A = 2i - 3j + 4k $ <br>Therefore, $ A_x = 2 $, $ A_y = -3 $, $ A_z = 4 $

Vector B: <br>$ B = i + 2j - k $ <br>Therefore, $ B_x = 1 $, $ B_y = 2 $, $ B_z = -1 $

Substituting these values into the dot product formula:

$ A \cdot B = (2 \times 1) + (-3 \times 2) + (4 \times -1) $

Calculating each term individually:

• $ 2 \times 1 = 2 $

• $ -3 \times 2 = -6 $

• $ 4 \times -1 = -4 $

Adding these results together:

$ 2 + (-6) + (-4) = 2 - 6 - 4 = -8 $

Therefore, the dot product A ⋅ B is -8.

The correct answer is -8. This is verified by following the dot product calculation method accurately and ensuring all component multiplications and additions are conducted correctly.

For more detailed understanding and examples on vector operations, you can refer to reputable academic sources such as:

• Wolfram MathWorld - Dot Product

• Khan Academy - Vector Dot Product and Magnitude