Direction Ratios of Resultant Vector?

📌 CONCEPT: The direction ratios of a vector are the coefficients of its unit vector representation, which can be obtained by dividing the vector by its magnitude.

📐 RULE / FORMULA: To find the direction ratios of the resultant vector, we first find the resultant vector by adding the given vectors, and then divide the resultant vector by its magnitude.

💡 WORKED EXAMPLE: Given vectors A = 2i - 3j + 4k and B = -3i + 2j - 5k, let's find the direction ratios of the resultant vector when A is multiplied by 2 and added to B. First, we find the resultant vector: R = 2A + B = 2(2i - 3j + 4k) + (-3i + 2j - 5k) = 1i - 4j + 3k. Then, we find the magnitude of R: |R| = √(12 + 42 + 32) = √7. Finally, we divide R by its magnitude to get the unit vector: R/|R| = (1/√7)i - (4/√7)j + (3/√7)k. The direction ratios of R are 1, -4, and 3.

⚠️ COMMON MISTAKE: Students often forget to divide the resultant vector by its magnitude to get the unit vector, which is necessary to obtain the correct direction ratios.