Distance between two points - Collinear

Chapter 11 Class 11 - Intro to Three Dimensional Geometry
Concept wise

### Transcript

Ex 11.2, 2 Show that points P (–2, 3, 5), Q (1, 2, 3) & R (7, 0, –1) are collinear. If three points are collinear, then they lie on a line. Let first calculate distance between the 3 points i.e. PQ. QR and PR Calculating PQ P ( – 2, 3, 5) Q (1, 2, 3) Hence , PQ = √((x2−x1)2+(y2−y1)2+(z2 −z1)2) PQ = √((1−(−2))2+(2−3)2+(3−5)2) = √((1+2)2+(2−3)2+(3−5)2) = √(32+(−1)2+(−2)2) = √(9+1+4) = √14 Calculating QR Q ( 1, 2, 3) R (7, 0, –1) QR = √((x2−x1)2+(y2−y1)2+(z2 −z1)2) Here x1 = 1, y1 = 2, z1 = 2 x2 = 7, y2 = 0, z2 = – 1 Here , x1 = – 2, y1 = 3, z1 = 5 x2 = 1, y2 = 2, z2 = 3 QR = √((7−1)2+(0−2)2+(−1−3)2) = √((6)2+(−2)2+(−4)2) = √(36+4+16) = √56 = √(14 × 2 × 2) = 2√14 Calculating PR P (–2, 3, 5), R (7, 0, –1) PR = √((x2−x1)2+(y2−y1)2+(z2 −z1)2) Here, x1 = – 2, y1 = 3, z1 = 5 x2 = 7, y2 = 0, z2 = – 1 PR = √((7−(−2))2+(0−3)2+(−1−5)2) = √((7+2)2+(−3)2+(−6)2) = √((9)2+9+36) = √(81+9+36) = √126 = √(14 × 3 × 3) = 𝟑√𝟏𝟒 Thus, PQ = √14 , QR = 2√14 & PR = 3√14 So, PQ + QR = √14 + 2√14 = 3√14 = PR Thus, PQ + QR = PR So, if we draw the points on a graph, with PQ + QR = PR We see that points P,Q,R lie on the same line. Thus, P, Q and R all collinear
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.teachoo.com/2797/1552/Ex-12.2--2---Show-that-points-P-(-2--3--5)--Q-(1--2--3)/category/Distance-between-two-points---Collinear/", "fetch_time": 1726237463000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-38/segments/1725700651523.40/warc/CC-MAIN-20240913133933-20240913163933-00318.warc.gz", "warc_record_offset": 947910288, "warc_record_length": 21751, "token_count": 694, "char_count": 1275, "score": 4.46875, "int_score": 4, "crawl": "CC-MAIN-2024-38", "snapshot_type": "latest", "language": "en", "language_score": 0.7167653441429138, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00041-of-00064.parquet"}