Question

solve this question ..

need proper solution .☺☺

Answer 1

Here, A, B denotes vector.
|A| ,|B| is magnitude of vector A and B respectively.

Q : 52 If A and B vectors denote the sides of a parallelogram and its area is |A|B|/2 .
Find angle between A and B.

Ans:52
The area of a parallelogram whose adjacent sides are A and B vectors is given by :
Area = | A x B.|
=> |A| |B| /2 = | A | |B| sin (a)
=> sin (a) = 1/2
a = 30° as a is acute angle.
Hence, angle between A and B is 30°.

Q : 53 Determine a unit vector perpendicular to A = 2i + j +k and B = i-j+2k .
Ans: 52
Vector perpendicular to A and B is given by
A x B.
Calculation of A x B is in pic 1.
A x B = 3(i -j-k)
| A x B | = 3√(1^2 + (-1)^2 + (-1)^2 )
= 3 √3


Unit Vector perpendicular to A and B is given by
=> A x B /(| A x B | )
=> 3(i - j - k) / 3√3 = (i-j-k) / √3 .

Q : 54 : Find a vector whose length is 7 and which is perpendicular to each of these vectors.
A = 2 i - 3j + 6k
B = i +j -k

Ans:54
Vector perpendicular to A and B is given by
A x B.
AxB = -3i + 8j + 5k
Let the required vector be k(-3i+8j+5k) whose magnitude is 7 .
=> k √(-3)^2+(8)^2 + (5)^2 ) = 7
=> k √9 + 64 +25 = 7
=> k √98 = 7
=> k *7 √2 = 7
=> k = 1/√2


Q : 55 Determine the sine of the angle between the vectors 3i+j+2k and 2i-2j +4k.
Ans:
We know that,
cos(a) = (3i+j+2k) . (2i-2j+4k) / (√24 √14 )
= 6-2+8 / (2√6 * √14)
= 12 / 2 √6 . √14
= √6/√14

sin a = √1 - (6/14)
. = 2 /.√7