The ratio in which `hat(i)+2hat(j)+3hat(k)` divides the join of `-2hat(i)+ 3hat(j)+5hat(k)` and `7hat(i)-hat(k)` is

The perimeter of the triangle with sides 3hat(i)+4hat(j)+5hat(k), 4hat(i)-3hat(j)+5hat(k) and 7hat(i)+hat(j) is

Show that the vectors vec(a), vec(b), vec(c) are coplanar, when (i) vec(a)=hat(i)-2hat(j)+3hat(k), vec(b) = -2hat(i)+3hat(j)-4hat(k) and vec(c)=hat(i)-3hat(j)+5hat(k) (ii) vec(a)=hat(i)+3hat(j)+hat(k), vec(b)=2hat(i)-hat(j)-hat(k)and vec(c)=7hat(j)+3hat(k) (iii) vec(a)=2hat(i)-hat(j)+2hat(k), vec(b)=hat(i)+2hat(j)-3hat(k) and vec(c)=3hat(i)-4hat(j)+7hat(k)

Find ( vec (a) xxvec (b)) and |vec(a) xx vec (b)| ,when (i) vec(a) = hat(i)-hat(j)+ 2hat(k) and vec(b)= 2 hat(i)+3 hat(j)-4hat(k) (ii) vec(a)= 2hat (i)+hat(j)+ 3hat(k) and vec(b)= 3hat(i)+5 hat(j) - 2 hat(k) (iii) vec(a)=hat(i)- 7 hat(j)+ 7hat(k) and vec(b) = 3 hat(i)-2hat(j)+2 hat(k) (iv) vec(a)= 4hat(i)+ hat(j)- 2hat(k) and vec(b) = 3 hat(i)+hat(k) (v) vec(a) = 3 hat(i) + 4 hat(j) and vec(b) = hat(i)+hat(j)+hat(k)

Show that the following points whose position vectors are given are collinear : (i) 5 hat(i) + 5 hat(k), 2 hat(i) + hat(j) + 3 hat(k) and - 4 hat(i) + 3 hat(j) - hat(k) (ii) - 2 hat(i) + 3 hat(j) + 5 hat(k), hat(i) + 2 hat(j) + 3 hat(k) and 7 hat(i) - hat(k) .

The two vectors A=2hat(i)+hat(j)+3hat(k) and B=7hat(i)-5hat(j)-3hat(k) are :-

Express -hat(i)-3hat(j)+4hat(k) as the linear combination of the vectors 2hat(i)+hat(j)-4hat(k) , 2hat(i)-hat(j)+3hat(k) and 3hat(i)+hat(j)-2hat(k) .

Find the angle between the vectors hat(i)+3hat(j)+7hat(k) and 7hat(i)-hat(j)+8hat(k) .

show that the points whose position vectors are (-2hat(i) +3hat(j) +5hat(k)) (hat(i)+2hat(j)+3hat(k)) " and " (7hat(i) -hat(k)) are collinear.

Find vec(A).vec(b) when (i) vec(a)=hat(i)-2hat(j)+hat(k) and vec(b)=3 hat(i)-4 hat(j)-2 hat(k) (ii) vec(a)=hat(i)+2hat(j)+3hat(k) and vec(b)=-2hat(j)+4hat(k) (iii) vec(a)=hat(i)-hat(j)+5hat(k) and vec(b)=3 hat(i)-2 hat(k)