10 త్రీ డైమెన్షనల్ జ్యామితి

వ్యాయామం 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

స్టడీ మెటీరియల్‌కి తిరిగి వెళ్ళు

JEE స్టడీ మెటీరియల్ (గణితం)

01 సంబంధాలు మరియు విధులు

02 విలోమ త్రికోణమితి విధులు

03 మాత్రికలు

04 నిర్ణాయకాలు

05 కొనసాగింపు మరియు భిన్నత్వం

06 డెరివేటివ్‌ల అప్లికేషన్

07 సమగ్రతలు

08 ఇంటిగ్రల్స్ యొక్క అప్లికేషన్

09 వెక్టర్స్

10 త్రీ డైమెన్షనల్ జ్యామితి

11 లీనియర్ ప్రోగ్రామింగ్

12 సంభావ్యత

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