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If l,m,n be three real numbers proportional to the direction cosines of a line L.then l²+m²+n²=1

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If a, b, c are three numbers proportional to the direction cosine l, m, n of a straight line, then a, b, c are called its direction ratios. They are also called direction numbers or direction components. Hence by definition, we have 1/a = m/b = n/c = k (say) => l=ak, m=bk, n=ck => k2(a2...
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If a, b, c are three numbers proportional to the direction cosine l, m, n of a straight line, then a, b, c are called its direction ratios. They are also called direction numbers or direction components. Hence by definition, we have 1/a = m/b = n/c = k (say) => l=ak, m=bk, n=ck => k2(a2 + b2 + c2) = l2 + m2 + n2=1 => k = ± 1 / ?a2 + b2 + c2 = ± 1/??a2 l = ± a/??a2. Similarly m = ± b/??a2 and n = ± n/??a2 where the same sign either positive or negative is to be chosen throughout. Example: If 2, – 3, 6 be the direction ratios, then the actual direction cosines are 2/7, –3/7, 6/7. Note: Direction cosines of a line are unique but direction ratios of a line in no way unique but can be infinite. read less
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Direction cosines can be written in terms of direction ratios. l = a/sqrt(a^2+b^2+c^2), m = b/sqrt(a^2+b^2+c^2) n = c/sqrt(a^2+b^2+c^2) So, l^2+m^2+n^2 = (a^2+b^2+c^2)/(a^2+b^2+c^2) =1
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If l , m and n are the direction cosines of any line, then l² + m² + n² = 1 Proof: Let OP be drawn through the origin parallel to the given line so that l , m , n are the cosines of the angles which OP makes with OX, OY and OZ respectively. Let (x, y , z) be the co-ordinates of any point P on this...
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If l , m and n are the direction cosines of any line, then l² + m² + n² = 1 Proof: Let OP be drawn through the origin parallel to the given line so that l , m , n are the cosines of the angles which OP makes with OX, OY and OZ respectively. Let (x, y , z) be the co-ordinates of any point P on this line. Let OP = r , then x = lr , y = mr , z = nr Squaring and adding , we obtain x² + y² + z² = ( l² + m² + n²) r² l² + m² + n² = (x² + y² + z²) / r² but r² = OP² =(x² + y² + z²) Thus l² + m² + n² = 1 Proved. read less
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RELATION BETWEEN DIRECTION COSINES: If l , m and n are the direction cosines of any line, then l² + m² + n² = 1 Proof: Let OP be drawn through the origin parallel to the given line so that l , m , n are the cosines of the angles which OP makes with OX, OY and OZ respectively. Let (x, y , z) be the...
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RELATION BETWEEN DIRECTION COSINES: If l , m and n are the direction cosines of any line, then l² + m² + n² = 1 Proof: Let OP be drawn through the origin parallel to the given line so that l , m , n are the cosines of the angles which OP makes with OX, OY and OZ respectively. Let (x, y , z) be the co-ordinates of any point P on this line. Let OP = r , then x = lr , y = mr , z = nr Squaring and adding , we obtain x² + y² + z² = ( l² + m² + n²) r² l² + m² + n² = (x² + y² + z²) / r² but r² = OP² =(x² + y² + z²) Thus l² + m² + n² = 1 Proved. read less
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