The vectors `bar X and bar Y` satisfy the equations `2bar X =bar p,bar X+2bar Y =bar q` where `bar p=bar i+bar j and bar q=bar i-bar j`. If `theta` is the angle between `bar X and bar Y` then

Let bar(a) = 2bar(i)-bar(j)+bar(k), bar(b) = 3bar(i) + 4bar(j) -bar(k) and if theta is the angle between bar(a), bar(b) then find sin theta .

If bar(a) = 2bar(i) + 2bar(j) - 3bar(k), bar(b) = 3bar(i) - bar(j) + 2bar(k) then find the angle between 2bar(a)+bar(b) and bar(a) + 2bar(b) .

If bar a = 2 bar I + bar j - bar k, bar b = - bari + 2 bar j - 4 bar k and bar c = bar I + bar j + bar k , then find ( bar a xx bar b) cdot (bar b xx bar c) .

bar (a) = 2bar (i) + 3bar (j) -bar (k), bar (b) = bar (i) + 2bar (j) -4bar (k), bar (c) = bar (i) + bar (j) + bar (k), bar (d) = bar (i) -bar (j) -bar (k) then

Find the vector area and area of the triangle with vertices bar(i) + bar(j)-bar(k), 2bar(i)- 3bar(j) + bar(k), 3 bar(i) + bar(j)- 2bar(k) .

If bar(r)=3bar(p)+4bar(q)" and "2bar(r)=bar(p)-3bar(q) then

Let bar(a)= bar(i) + 2bar(j) + 3bar(k), bar(b)= - bar(i) + 2bar(j) + 3bar(k), bar( c)= 3 bar(i) + bar(j) and bar(d) is a non-zero vector perpendicular to both bar(a) and bar(b) . If the angle between bar(c ) and bar(d) is theta , then find cos theta .

For three vectors bar(p), bar(q) and bar(r)" if "bar(r)=3bar(p)+4bar(q) and 2bar(r)=bar(p)-3bar(q) then

If bar(a) = 2bar(i) - bar(j) + bar(k), and bar(b) = bar(i) - 3bar(j) - 5bar(k) then find |bar(a) xx bar(b)| .

A non-zero vector bar(a) , is parallel to the line of intersection of the plane determined by the vectors bar(i), bar(i) + bar(j) and the plane determined by the vectors bar(i)- bar(j), bar(i) + bar(k) Find the angle between bar(a) and the vector bar(i)- 2bar(j) + 2bar(k)