Which of the following vector is perpendicular to the vector A= 2i + 3j + 4k:
Correct answer: D. i+2j-2k
- A. i+j+k
- B. 4i+3j-2k
- C. i-3j+k
- D. i+2j-2k
Explanation
To determine if a vector is perpendicular to another vector, we can use the dot product. If the dot product of two vectors is zero, they are perpendicular to each other. Let's calculate the dot product of vector A (2i + 3j + 4k) and each of the given options: A) A · (i + j + k) = (2i + 3j + 4k) · (i + j + k) = 2 + 3 + 4 = 9 (not zero) B) A · (4i + 3j - 2k) = (2i + 3j + 4k) · (4i + 3j - 2k) = 8 + 9 - 8 = 9 (not zero) C) A · (i - 3j + k) = (2i + 3j + 4k) · (i - 3j + k) = 2 - 9 + 4 = -3 (not zero) D) A · (i + 2j - 2k) = (2i + 3j + 4k) · (i + 2j - 2k) = 2 + 6 - 8 = 0 The dot product of vector A and option D is zero, indicating that they are perpendicular. Therefore, option D) i + 2j - 2k is the vector that is perpendicular to vector A.
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About Scalar Product
The scalar product of two vectors is A dot B equals AB cos theta, producing a scalar determined by their magnitudes and included angle. Applications include work done and projection of one vector on another, with perpendicular vectors giving zero and the scalar product distinguished from the vector product.
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