Show that the line joining the points P and Q with position vectors `vec(p)=p_(1)hat(i)+p_(2)hat(j)+p_(3)hat(k)` and `vec(q)hat(i)+q_(2)hat(j)+q_(3)hat(k)` passes through the origin if `vec(p)*vec(q)=|vec(p)||vec(q)|`.

If vectors vec(alpha)=2hat(i)+3hat(j)-6hat(k) " and " vec(beta)=phat(i)-hat(j)+2hat(k) are parallel, then the value of p is -

Find a unit vector perpendicular to both the vector vec(a) = 2 hat(i) + hat(j) - 2 hat(k) and vec(b) = 3 hat (i) - hat (j) + hat (k)

If vectors vec (a) = p hat (i) + 8 hat (j) + 6 hat (k) and vec (b) = - 3 hat (i) + 4 hat (j) + q hat (k) are parallel , find p and q

vec(a)=6hat(i)+7hat(j)+7hat(k),vec(b)=3hat(i)+2hat(j)-2hat(k),P(1,2,3) The image of the points P in the line vec(r)=vec(a)+lamdavec(b) is -

If vec(a)=2hat(i)+3hat(j)-4hat(k) " and " vec(b)=hat(i)+2hat(j)+hat(k) , then find (vec(a)+vec(b)) " and " abs(vec(a)+vec(b)).

If vec (a)= 2 hat (i) - hat (j) + hat (k) and vec(b) = - hat (i) + 3 hat (j) + 4 hat (k) , then value of vec (a). vec(b) is -

If hat (i),hat (j) and hat (k) are unit vectors along three mutuaaly perpendicular axes and vec (a) = a_(1) hat (i) + a_(2) hat(j) + a_(3) hat (k) , vec(b) = b_(1) hat (i) + b_(2) hat (j) + b_(3) hat (k) and vec ( c ) = c_(1) hat(i) + c_(2) hat (j) + c_(3) hat (k) prove that vec(a) . (vec(b)+vec(c)) = vec(a).vec(b) + vec(a).vec(c)

If vec(a)=2hat(i)-5hat(j)+3hat(k) " and " vec(b)=hat(i)-2hat(j)-4hat(k) , find the value of abs(3vec(a)+2vec(b)).

If vec(a) = hat (i) +hat (j) , vec (b) = hat(i) - hat(j) and vec(c ) = 5 hat (i) + 2 hat (j) + 3 hat(k) , find the value of [ vec(b)vec (c ) vec(a)]

If vec(a)=hat(i)+2hat(j)-hat(k) " and " vec(b)=3hat(i)+hat(j)-5hat(k) , find a unit vector in a direction parallel to vector (vec(a)-vec(b)) .