Mathematics : Applications of Vector Algebra: Application of Vectors to 3-Dimensional Geometry: Angle between two straight lines

The acute angle between two given straight lines

If two lines are given in Cartesian form as then the acute angle Î¸ between the two given lines is given by

Remark

(i) The two given lines with direction ratios b1 , b2 , b3 and d1 , d2 , d3 are parallel if, and only if .

(ii) The two given lines with direction ratios b1, b2, b3 and d1, d2, d3 are perpendicular if and only if *b*1*d*1 + *b*2*d*2 + *b*3*d*3 = 0 .

(iii) If the direction cosines of two given straight lines are l1 , m1 , n1 and l2 , m2 , n2, then the angle between the two given straight lines is cos Î¸ =| l1l2 + m1m2 + n1n2 | .

Find the acute angle between the lines * *= (Ë†*i *+ 2Ë†*j *+ 4Ë†*k *) + *t*(2Ë†*i *+ 2Ë†*j *+ Ë†*k *) and the straight line passing through the points (5,1, 4) and (9, 2,12) .

We know that the line * *= (Ë†*i *+ 2Ë†*j *+ 4Ë†*k *) + *t*(2Ë†*i *+ 2Ë†*j *+ Ë†*k *) is parallel to the vector 2Ë†*i *+ 2Ë†*j *+ Ë†*k*.

Direction ratios of the straight line joining the two given points (5,1, 4) and (9, 2,12) are 4,1,8 and hence this line is parallel to the vector 4*i*Ë† + Ë†*j *+ 8*k*Ë† .

Therefore, the acute angle between the given two straight lines is

Find the acute angle between the straight lines and state whether they are parallel or perpendicular.

Comparing the given lines with the general Cartesian equations of straight lines,

we find (*b*1 , *b*2 , *b*3 ) = (2,1, âˆ’2) and (*d*1 , *d*2 , *d*3 ) = (4, âˆ’4, 2) . Therefore, the acute angle between the two straight lines is

Thus the two straight lines are perpendicular.

Show that the straight line passing through the points *A*(6, 7, 5) and *B*(8,10, 6) is perpendicular to the straight line passing through the points *C*(10, 2, âˆ’5) and *D*(8, 3, âˆ’4) .

The straight line passing through the points* A*(6, 7, 5) and *B*(8,10, 6) is parallel to the vector * *= * *= * *âˆ’ * *= 2*i*Ë† + 3 Ë†*j *+ *k*Ë† and the straight line passing through the points *C*(10, 2, âˆ’5) and *D*(8, 3, âˆ’4) is parallel to the vector * *= * *= âˆ’2*i*Ë† + Ë†*j *+ *k*Ë† . Therefore, the angle between the two straight lines is the angle between the two vectors * *and . Since

the two vectors are perpendicular, and hence the two straight lines are perpendicular.

Aliter

We find that direction ratios of the straight line joining the points *A*(6, 7, 5) and *B*(8,10, 6) are (*b*1 , *b*2 , *b*3 ) = (2, 3,1) and and direction ratios of the line joining the points *C*(10, 2, âˆ’5) and *D*(8, 3, âˆ’4) are (*d*1 , *d*2 , *d*3 ) = (âˆ’2,1,1) . Since *b*1*d*1 + *b*2*d*2 + *b*3*d*3 = (2)(âˆ’2) + (3)(1) + (1)(1) = 0 , the two straight lines are perpendicular.

Example 6.32

Show that the lines and are parallel

*Solution*

We observe that the straight line is parallel to the vector 4*i*Ë† - 6 Ë†*j* +12*k*Ë† and the straight line is parallel to the vector -2*i*Ë† + 3Ë†*j* - 6*k*Ë†.

Since 4*i*Ë† - 6Ë†*j* +12*k*Ë† = -2(-2*i*Ë† + 3Ë†*j* - 6*k*Ë†) , the two vectors are parallel, and hence the two straight lines are parallel.

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Angle between two straight lines | Definition, Theorem, Proof, Solved Example Problems, Solution

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