10 त्रिविमीय ज्यामिति का परिचय
अभ्यास 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
जेईई अध्ययन सामग्री (गणित)
01 संबंध एवं फलन
02 व्युत्क्रम त्रिकोणमितीय फलन
03 आव्यूह
04 सारणिक
05 सांत्यता और अवकलनीयता
06 अवकलज का अनुप्रयोग
07 समाकलन
08 समाकलन का अनुप्रयोग
09 वैक्टर
10 त्रिविमीय ज्यामिति का परिचय
11 रैखिक प्रोग्रामिंग
12 प्रायिकता