10 ତିନୋଟି ଡାଇମେନ୍ସନାଲ୍ ଜ୍ୟାମିତି

ବ୍ୟାୟାମ 01

Question:

If a line has the direction ratios −18,12,−4, then what are its direction cosines?

Answer:

Answer: Step 1: Direction ratios are the components of a vector that are expressed as a fraction of the magnitude of the vector.

Step 2: Direction cosines of a vector are the cosines of the angles between the vector and the three coordinate axes.

Step 3: To find the direction cosines of the line, we need to find the angles between the line and the coordinate axes.

Step 4: The angle between the line and the x-axis is given by tanθx = 12/−18 = −2/3.

Step 5: The angle between the line and the y-axis is given by tanθy = −4/12 = −1/3.

Step 6: The angle between the line and the z-axis is given by tanθz = 12/−4 = −3.

Step 7: The direction cosines of the line are cosθx = −2/3, cosθy = −1/3 and cosθz = −3.

Question:

If a line makes angles 90∘,135∘,45∘ with the x,y and Z−axes respectively, find its direction cosines.

Answer:

Answer:

Step 1: The line makes an angle of 90∘ with the x-axis.

Step 2: Therefore, the direction cosine of the line along the x-axis (lx) is 1.

Step 3: The line makes an angle of 135∘ with the y-axis.

Step 4: Therefore, the direction cosine of the line along the y-axis (ly) is -√2/2.

Step 5: The line makes an angle of 45∘ with the z-axis.

Step 6: Therefore, the direction cosine of the line along the z-axis (lz) is √2/2.

Therefore, the direction cosines of the line are lx = 1, ly = -√2/2 and lz = √2/2.

Question:

Find the direction cosines of a line which makes equal angles with the coordinate axes.

Answer:

Step 1: A line which makes equal angles with the coordinate axes is at an angle of 45° with the x-axis and y-axis.

Step 2: The direction cosines of a line at an angle of 45° with the x-axis and y-axis can be calculated using the formula: cosθx = cos45° = √2/2 cosθy = cos45° = √2/2

Step 3: Therefore, the direction cosines of the line which makes equal angles with the coordinate axes are: cosθx = √2/2 cosθy = √2/2

ଅଧ୍ୟୟନ ସାମଗ୍ରୀକୁ ଫେରନ୍ତୁ

JEE ଅଧ୍ୟୟନ ସାମଗ୍ରୀ (ଗଣିତ)

01 ସମ୍ପର୍କ ଏବଂ କାର୍ଯ୍ୟ

02 ଓଲଟା ଟ୍ରାଇଗୋନେଟ୍ରିକ୍ କାର୍ଯ୍ୟଗୁଡ଼ିକ

03 ମ୍ୟାଟ୍ରିକ୍ସ

04 ନିର୍ଣ୍ଣୟକାରୀ

05 ନିରନ୍ତରତା ଏବଂ ଭିନ୍ନତା

06 ଡେରିଭେଟିକ୍ସର ପ୍ରୟୋଗ

07 ଇଣ୍ଟିଗ୍ରାଲ୍

08 ଇଣ୍ଟିଗ୍ରାଲ୍ସର ପ୍ରୟୋଗ

09 ଭେକ୍ଟର୍

10 ତିନୋଟି ଡାଇମେନ୍ସନାଲ୍ ଜ୍ୟାମିତି

11 ରେଖା ପ୍ରୋଗ୍ରାମିଂ

12 ସମ୍ଭାବନା

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