{"id":3918,"date":"2021-08-16T20:46:09","date_gmt":"2021-08-16T20:46:09","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3918"},"modified":"2021-10-10T20:00:14","modified_gmt":"2021-10-10T14:30:14","slug":"definition-of-collinear-vectors","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/definition-of-collinear-vectors\/","title":{"rendered":"Definition of Collinear Vectors"},"content":{"rendered":"
Here, you will learn definition of collinear vectors, coplanar vectors, co-initial vectors and test of collinearity of three points.<\/p>\n
Let’s begin –<\/p>\n
Two vectors are said to be collinear if their supports are parallel disregards to their direction. Collinear vectors are also called Parallel vectors<\/strong>. If they have the same direction they are named as like vectors otherwise unlike vectors.<\/p>\n Symbolically, If \\(\\vec{a}\\) & \\(\\vec{b}\\) are collinear or parallel vectors, then there exists a scalar \\(\\lambda\\) such that \\(\\vec{a}\\) = \\(\\lambda\\vec{b}\\) or, \\(\\vec{b}\\) = \\(\\lambda\\vec{a}\\).<\/p>\n<\/blockquote>\n Theorem 1 :<\/b><\/p>\n Two non-zero vectors \\(\\vec{a}\\) & \\(\\vec{b}\\) are collinear iff there exist scalars x, y not both zero such that x\\(\\vec{a}\\) + y\\(\\vec{b}\\) = \\(\\vec{0}\\).<\/p>\n<\/blockquote>\n Theorem 2 :<\/b><\/p>\n If \\(\\vec{a}\\) & \\(\\vec{b}\\) are two non-zero non-collinear vectors and x, y are scalars then x\\(\\vec{a}\\) + y\\(\\vec{b}\\) = 0 \\(\\implies\\) x = y = 0.<\/p>\n<\/blockquote>\n Example<\/strong><\/span> : If \\(\\vec{a}\\) and \\(\\vec{b}\\) are non-collinear vectors, find the value of x for which vectors \\(\\vec{\\alpha}\\) = (x – 2)\\(\\vec{a}\\) + \\(\\vec{b}\\) and \\(\\vec{\\beta}\\) = (3 + 2x)\\(\\vec{a}\\) – 2\\(\\vec{b}\\) are collinear.<\/p>\n Solution<\/span><\/strong> : Since vectors \\(\\vec{\\alpha}\\) and \\(\\vec{\\beta}\\) are collinear. Therefore, there exist scalar \\(\\lambda\\) such that<\/p>\n \\(\\vec{\\alpha}\\) = \\(\\lambda\\)\\(\\vec{\\beta}\\)<\/p>\n \\(\\implies\\) (x – 2)\\(\\vec{a}\\) + \\(\\vec{b}\\) = \\(\\lambda\\){(3 + 2x)\\(\\vec{a}\\) – 2\\(\\vec{b}\\)}<\/p>\n \\(\\implies\\) {x – 2 – \\(\\lambda\\)(3 + 2x)}\\(\\vec{a}\\) + (1 + 2\\(\\lambda\\)\\(\\vec{b}\\) = \\(\\vec{0}\\)<\/p>\n Now, given \\(\\vec{a}\\) and \\(\\vec{b}\\) are non-collinear.<\/p>\n Therefore, from theorem 2,<\/p>\n x – 2 – \\(\\lambda\\)(3 + 2x) = 0 and 1 + 2\\(\\lambda\\) = 0<\/p>\n x – 2 – \\(\\lambda\\)(3 + 2x) = 0 and \\(\\lambda\\) = \\(-1\\over 2\\)<\/p>\n x – 2 + \\(1\\over 2\\)(3 + 2x) = 0 \\(\\implies\\) 4x + 1 = 0<\/p>\n \\(\\implies\\) x = \\(-1\\over 4\\)<\/p>\n (a)\u00a0 3 points A B C will be collinear if \\(\\overrightarrow{AB}\\) = \\(\\lambda\\overrightarrow{BC}\\), where \\(\\lambda\\) \\(\\in\\) R.<\/p>\n (b)\u00a0 Three points A, B, C with position vectors \\(\\vec{a}\\),\\(\\vec{b}\\),\\(\\vec{c}\\) respectively are collinear, if & only if there exist scalars x,y,z not all zero simultaneously such that ; x\\(\\vec{a}\\) + y\\(\\vec{b}\\) + z\\(\\vec{c}\\) = 0, where x + y + z = 0.<\/p>\n (c)\u00a0 Collinearity can also be checked by first finding the equation of line through two points and satisfy the third point.<\/p>\n\n\n\n
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Test of Collinearity of three points in Vectors<\/h2>\n