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If the volume of a parallelepiped whose coterminous edges are given by the vectors vec a=hat i+hat j+nhat k vec b=2i+4j-nhat k and vec c=hat i+hat j+3hat k(n>=0) is 158 cu.Units then |
| Answer» <html><body><p><strong>Answer:</strong></p><p></p><p>Search for questions, <a href="https://interviewquestions.tuteehub.com/tag/posts-25660" style="font-weight:bold;" target="_blank" title="Click to know more about POSTS">POSTS</a> and chapters</p><p>Maths</p><p>Bookmark</p><p>Let the <a href="https://interviewquestions.tuteehub.com/tag/volume-728707" style="font-weight:bold;" target="_blank" title="Click to know more about VOLUME">VOLUME</a> of a parallelopiped whose coterminous edges are given by </p><p>u</p><p> = </p><p>i</p><p>^</p><p> + </p><p>j</p><p>^</p><p> </p><p> +λ </p><p><a href="https://interviewquestions.tuteehub.com/tag/k-527196" style="font-weight:bold;" target="_blank" title="Click to know more about K">K</a></p><p>^</p><p> , </p><p>v</p><p> = </p><p>i</p><p>^</p><p> + </p><p>j</p><p>^</p><p> </p><p> +3 </p><p>k</p><p>^</p><p> and</p><p>w</p><p> =2 </p><p>i</p><p>^</p><p> + </p><p>j</p><p>^</p><p> </p><p> + </p><p>k</p><p>^</p><p> be 1cu. <a href="https://interviewquestions.tuteehub.com/tag/units-1438468" style="font-weight:bold;" target="_blank" title="Click to know more about UNITS">UNITS</a>. If θ be the angle between the edge </p><p>u</p><p> and </p><p>w</p><p> , then cosθ can be: </p><p> </p><p> </p><p> </p><p> </p><p> </p><p> </p><p> </p><p>share</p><p>Share</p><p>Answer</p><p>Correct option is</p><p>D</p><p>6 </p><p>3</p><p> </p><p> </p><p>7</p><p> </p><p> </p><p>Given, </p><p>u</p><p> = </p><p>i</p><p>^</p><p> + </p><p>j</p><p>^</p><p> </p><p> +λ </p><p>k</p><p>^</p><p> ,</p><p>v</p><p> = </p><p>i</p><p>^</p><p> + </p><p>j</p><p>^</p><p> </p><p> +3 </p><p>k</p><p>^</p><p> </p><p>w</p><p> =2 </p><p>i</p><p>^</p><p> + </p><p>j</p><p>^</p><p> </p><p> + </p><p>k</p><p>^</p><p> </p><p></p><p>Volume of parallelopied =u.( </p><p>v</p><p> × </p><p>w</p><p> )=[ </p><p>u</p><p> </p><p>v</p><p> </p><p>w</p><p> ]</p><p>= </p><p>∣</p><p>∣</p><p>∣</p><p>∣</p><p>∣</p><p>∣</p><p>∣</p><p>∣</p><p> </p><p> </p><p>1</p><p>1</p><p>2</p><p> </p><p> </p><p>1</p><p>1</p><p>1</p><p> </p><p> </p><p>λ</p><p>λ</p><p>1</p><p> </p><p> </p><p>∣</p><p>∣</p><p>∣</p><p>∣</p><p>∣</p><p>∣</p><p>∣</p><p>∣</p><p> </p><p> </p><p></p><p>=1(1−3)−1(1−6)+λ(1−2)−2+5+(−λ)3−λ</p><p></p><p>so,</p><p>Given volume =±1 taking (+) sign</p><p>3−λ=1</p><p>λ=2</p><p></p><p>Angle between </p><p>u</p><p> and </p><p>w</p><p> </p><p></p><p>cosθ= </p><p>∣ </p><p>u</p><p> ∣.∣ </p><p>w</p><p> ∣</p><p>u</p><p> . </p><p>w</p><p> </p><p> </p><p> = </p><p>1+1+4</p><p> </p><p> . </p><p>4+1+1</p><p> </p><p> </p><p>(i+j+2k).(2i+j+k)</p><p> </p><p> </p><p></p><p>= </p><p>5</p><p> </p><p> . </p><p>5</p><p> </p><p> </p><p>2+1+2</p><p> </p><p> </p><p></p><p>= </p><p>5</p><p>5</p><p> </p><p> =1</p><p></p><p>Taking (−) sign</p><p>3−λ=−1</p><p></p><p>λ=4</p><p></p><p>u</p><p> = </p><p>i</p><p>^</p><p> + </p><p>j</p><p>^</p><p> </p><p> +4 </p><p>k</p><p>^</p><p> </p><p></p><p>w</p><p> =2i+j+ </p><p>k</p><p>^</p><p> </p><p></p><p>cosθ= </p><p>∣ </p><p>u</p><p> ∣∣ </p><p>w</p><p> ∣</p><p>u</p><p> . </p><p>w</p><p> </p><p> </p><p> = </p><p>1+1+16</p><p> </p><p> </p><p>4+1+1</p><p> </p><p> </p><p>( </p><p>i</p><p>^</p><p> + </p><p>j</p><p>^</p><p> </p><p> +4 </p><p>k</p><p>^</p><p> ).(2i+j+k)</p><p> </p><p> </p><p></p><p>= </p><p>18</p><p> </p><p> </p><p>6</p><p> </p><p> </p><p>2+1+4</p><p> </p><p> = </p><p>6 </p><p>3</p><p> ....Answer</p></body></html> | |