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Walchand College of Engineering (WCE), Sangli
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## Angle between two lines given direction ratios of two lines Solution

STEP 0: Pre-Calculation Summary
Formula Used
angle_b = acos(((Direction ratio 1 of line1*Direction ratio 1 of line2)+(Direction ratio 2 of line1*Direction ratio 2 of line2)+(Direction ratio 3 of line1*Direction ratio 3 of line2))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction ratio 1 of line2)^2+(Direction ratio 2 of line2)^2+(Direction ratio 3 of line2)^2)))
∠B = acos(((a1_line1*a1_line2)+(b2_line1*b2_line2)+(c3_line1*c3_line2))/(sqrt((a1_line1)^2+(b2_line1)^2+(c3_line1)^2))*(sqrt((a1_line2)^2+(b2_line2)^2+(c3_line2)^2)))
This formula uses 3 Functions, 6 Variables
Functions Used
cos - Trigonometric cosine function, cos(Angle)
acos - Inverse trigonometric cosine function, acos(Number)
sqrt - Squre root function, sqrt(Number)
Variables Used
Direction ratio 1 of line1 - Direction ratio 1 of line1 is defined as the quotient of two x coordinates of line1. (Measured in Hundred)
Direction ratio 1 of line2 - Direction ratio 1 of line2 is defined as the quotient of two x coordinates of line 2. (Measured in Hundred)
Direction ratio 2 of line1 - Direction ratio 2 of line1 is defined as the quotient of two y coordinates of line 1. (Measured in Hundred)
Direction ratio 2 of line2 - Direction ratio 2 of line2 is defined as the quotient of two y coordinates of line 2. (Measured in Hundred)
Direction ratio 3 of line1 - Direction ratio 3 of line1 is defined as the quotient of two z coordinates of line 1. (Measured in Hundred)
Direction ratio 3 of line2 - Direction ratio 3 of line2 is defined as the quotient of two z coordinates of line 2. (Measured in Hundred)
STEP 1: Convert Input(s) to Base Unit
Direction ratio 1 of line1: 0.2 Hundred --> 0.2 Hundred No Conversion Required
Direction ratio 1 of line2: 0.4 Hundred --> 0.4 Hundred No Conversion Required
Direction ratio 2 of line1: 0.5 Hundred --> 0.5 Hundred No Conversion Required
Direction ratio 2 of line2: 0.2 Hundred --> 0.2 Hundred No Conversion Required
Direction ratio 3 of line1: 0.7 Hundred --> 0.7 Hundred No Conversion Required
Direction ratio 3 of line2: 0.44 Hundred --> 0.44 Hundred No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
∠B = acos(((a1_line1*a1_line2)+(b2_line1*b2_line2)+(c3_line1*c3_line2))/(sqrt((a1_line1)^2+(b2_line1)^2+(c3_line1)^2))*(sqrt((a1_line2)^2+(b2_line2)^2+(c3_line2)^2))) --> acos(((0.2*0.4)+(0.5*0.2)+(0.7*0.44))/(sqrt((0.2)^2+(0.5)^2+(0.7)^2))*(sqrt((0.4)^2+(0.2)^2+(0.44)^2)))
Evaluating ... ...
∠B = 1.21679149974547
STEP 3: Convert Result to Output's Unit
1.21679149974547 Radian -->69.7170174828223 Degree (Check conversion here)
69.7170174828223 Degree <-- Angle B
(Calculation completed in 00.015 seconds)

## < 4 Angle in 3D space Calculators

Angle between two planes
angle_c = acos(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to x axis)+(Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to y axis)+(Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to z axis))/(sqrt((Direction cosine 1 with respect to x axis)^2+(Direction cosine 1 with respect to y axis)^2+(Direction cosine 1 with respect to z axis)^2)* sqrt((Direction cosine 2 with respect to x axis)^2+(Direction cosine 2 with respect to y axis)^2+(Direction cosine 2 with respect to z axis)^2))) Go
Angle made by direction cosines of two lines in sine form
angle = asin(sqrt(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to y axis)- (Direction cosine 2 with respect to x axis*Direction cosine 1 with respect to y axis))^2+((Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to z axis)-(Direction cosine 2 with respect to y axis*Direction cosine 1 with respect to z axis))^2+((Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to x axis)-(Direction cosine 2 with respect to z axis*Direction cosine 1 with respect to x axis))^2)) Go
Angle between two lines given direction ratios of two lines
angle_b = acos(((Direction ratio 1 of line1*Direction ratio 1 of line2)+(Direction ratio 2 of line1*Direction ratio 2 of line2)+(Direction ratio 3 of line1*Direction ratio 3 of line2))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction ratio 1 of line2)^2+(Direction ratio 2 of line2)^2+(Direction ratio 3 of line2)^2))) Go
Angle between line and plane given coefficients of line and plane
angle_a = asin(((Direction ratio 1 of line1*Direction Ratio 1)+(Direction ratio 2 of line1*Direction Ratio 2)+(Direction ratio 3 of line1*Direction Ratio 3))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction Ratio 1)^2+(Direction Ratio 2)^2+(Direction Ratio 3)^2))) Go

### Angle between two lines given direction ratios of two lines Formula

angle_b = acos(((Direction ratio 1 of line1*Direction ratio 1 of line2)+(Direction ratio 2 of line1*Direction ratio 2 of line2)+(Direction ratio 3 of line1*Direction ratio 3 of line2))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction ratio 1 of line2)^2+(Direction ratio 2 of line2)^2+(Direction ratio 3 of line2)^2)))
∠B = acos(((a1_line1*a1_line2)+(b2_line1*b2_line2)+(c3_line1*c3_line2))/(sqrt((a1_line1)^2+(b2_line1)^2+(c3_line1)^2))*(sqrt((a1_line2)^2+(b2_line2)^2+(c3_line2)^2)))

## What is coordinate system in 3D space?

The three mutually perpendicular lines in a space which divides the space into eight parts and if these perpendicular lines are the coordinate axes, then it is said to be a coordinate system.

## How to Calculate Angle between two lines given direction ratios of two lines?

Angle between two lines given direction ratios of two lines calculator uses angle_b = acos(((Direction ratio 1 of line1*Direction ratio 1 of line2)+(Direction ratio 2 of line1*Direction ratio 2 of line2)+(Direction ratio 3 of line1*Direction ratio 3 of line2))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction ratio 1 of line2)^2+(Direction ratio 2 of line2)^2+(Direction ratio 3 of line2)^2))) to calculate the Angle B, Angle between two lines given direction ratios of two lines is defined as the angle formed by two lines corresponding to given direction ratios of two lines. Angle B and is denoted by ∠B symbol.

How to calculate Angle between two lines given direction ratios of two lines using this online calculator? To use this online calculator for Angle between two lines given direction ratios of two lines, enter Direction ratio 1 of line1 (a1_line1), Direction ratio 1 of line2 (a1_line2), Direction ratio 2 of line1 (b2_line1), Direction ratio 2 of line2 (b2_line2), Direction ratio 3 of line1 (c3_line1) & Direction ratio 3 of line2 (c3_line2) and hit the calculate button. Here is how the Angle between two lines given direction ratios of two lines calculation can be explained with given input values -> 69.71702 = acos(((0.2*0.4)+(0.5*0.2)+(0.7*0.44))/(sqrt((0.2)^2+(0.5)^2+(0.7)^2))*(sqrt((0.4)^2+(0.2)^2+(0.44)^2))).

### FAQ

What is Angle between two lines given direction ratios of two lines?
Angle between two lines given direction ratios of two lines is defined as the angle formed by two lines corresponding to given direction ratios of two lines and is represented as ∠B = acos(((a1_line1*a1_line2)+(b2_line1*b2_line2)+(c3_line1*c3_line2))/(sqrt((a1_line1)^2+(b2_line1)^2+(c3_line1)^2))*(sqrt((a1_line2)^2+(b2_line2)^2+(c3_line2)^2))) or angle_b = acos(((Direction ratio 1 of line1*Direction ratio 1 of line2)+(Direction ratio 2 of line1*Direction ratio 2 of line2)+(Direction ratio 3 of line1*Direction ratio 3 of line2))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction ratio 1 of line2)^2+(Direction ratio 2 of line2)^2+(Direction ratio 3 of line2)^2))). Direction ratio 1 of line1 is defined as the quotient of two x coordinates of line1, Direction ratio 1 of line2 is defined as the quotient of two x coordinates of line 2, Direction ratio 2 of line1 is defined as the quotient of two y coordinates of line 1, Direction ratio 2 of line2 is defined as the quotient of two y coordinates of line 2, Direction ratio 3 of line1 is defined as the quotient of two z coordinates of line 1 & Direction ratio 3 of line2 is defined as the quotient of two z coordinates of line 2.
How to calculate Angle between two lines given direction ratios of two lines?
Angle between two lines given direction ratios of two lines is defined as the angle formed by two lines corresponding to given direction ratios of two lines is calculated using angle_b = acos(((Direction ratio 1 of line1*Direction ratio 1 of line2)+(Direction ratio 2 of line1*Direction ratio 2 of line2)+(Direction ratio 3 of line1*Direction ratio 3 of line2))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction ratio 1 of line2)^2+(Direction ratio 2 of line2)^2+(Direction ratio 3 of line2)^2))). To calculate Angle between two lines given direction ratios of two lines, you need Direction ratio 1 of line1 (a1_line1), Direction ratio 1 of line2 (a1_line2), Direction ratio 2 of line1 (b2_line1), Direction ratio 2 of line2 (b2_line2), Direction ratio 3 of line1 (c3_line1) & Direction ratio 3 of line2 (c3_line2). With our tool, you need to enter the respective value for Direction ratio 1 of line1, Direction ratio 1 of line2, Direction ratio 2 of line1, Direction ratio 2 of line2, Direction ratio 3 of line1 & Direction ratio 3 of line2 and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
How many ways are there to calculate Angle B?
In this formula, Angle B uses Direction ratio 1 of line1, Direction ratio 1 of line2, Direction ratio 2 of line1, Direction ratio 2 of line2, Direction ratio 3 of line1 & Direction ratio 3 of line2. We can use 4 other way(s) to calculate the same, which is/are as follows -
• angle = asin(sqrt(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to y axis)- (Direction cosine 2 with respect to x axis*Direction cosine 1 with respect to y axis))^2+((Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to z axis)-(Direction cosine 2 with respect to y axis*Direction cosine 1 with respect to z axis))^2+((Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to x axis)-(Direction cosine 2 with respect to z axis*Direction cosine 1 with respect to x axis))^2))
• angle_b = acos(((Direction ratio 1 of line1*Direction ratio 1 of line2)+(Direction ratio 2 of line1*Direction ratio 2 of line2)+(Direction ratio 3 of line1*Direction ratio 3 of line2))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction ratio 1 of line2)^2+(Direction ratio 2 of line2)^2+(Direction ratio 3 of line2)^2)))
• angle_a = asin(((Direction ratio 1 of line1*Direction Ratio 1)+(Direction ratio 2 of line1*Direction Ratio 2)+(Direction ratio 3 of line1*Direction Ratio 3))/(sqrt((Direction ratio 1 of line1)^2+(Direction ratio 2 of line1)^2+(Direction ratio 3 of line1)^2))*(sqrt((Direction Ratio 1)^2+(Direction Ratio 2)^2+(Direction Ratio 3)^2)))
• angle_c = acos(((Direction cosine 1 with respect to x axis*Direction cosine 2 with respect to x axis)+(Direction cosine 1 with respect to y axis*Direction cosine 2 with respect to y axis)+(Direction cosine 1 with respect to z axis*Direction cosine 2 with respect to z axis))/(sqrt((Direction cosine 1 with respect to x axis)^2+(Direction cosine 1 with respect to y axis)^2+(Direction cosine 1 with respect to z axis)^2)* sqrt((Direction cosine 2 with respect to x axis)^2+(Direction cosine 2 with respect to y axis)^2+(Direction cosine 2 with respect to z axis)^2))) Let Others Know