Show that the vector `hat(i)+hat(j)+hat(k)` makes the same angle with the positive directions of coordinates axes.

Show that the vectors -hat(i)+hat(j), -4hat(i)-6hat(j) " and " 5hat(i)+5hat(j) are the sides of a right-angled triangle.

The position vector of the points P and Q are 5 hat i+7 hat j-2 hat k and -3 hat i+3 hat j+6 hat k , respectively. Vector vec A=3 hat i- hat j+ hat k passes through point P and vector vec B=-3 hat i+2 hat j+4 hat k passes through point Q . A third vector 2 hat i+7 hat j-5 hat k intersects vectors A and Bdot Find the position vectors of points of intersection.

The scaler product of the vector hat (i) + hat (j) + hat (k) with the unit vector along the sum of vectors 2 hat (i) + 4 hat (j) - 5 hat (k) and lambda hat (i) + 2 hat (j) + 3 hat (k) is equal to one . Find the value of lambda

Find the vector equation of a line which passes through the point with position vector hat(i)-2hat(j)+4hat(k) and is in the direction of hat(i)+2hat(j)-hat(k) . Also, reduce it to cartesian from.

If vec(a)=hat(i)+hat(j)-4hat(k) " and " vec(b)=4hat(i)-hat(j)-2hat(k) , then find (i) a unit vector in the direction of the vector (2vec(a)-vec(b)) and (ii) vector and scalar components of the vector (2vec(a)-vec(b)) along coordinate axes.

Show that the lines vec(r)=(hat(i)+hat(j)+hat(k))+t(hat(i)-hat(j)+hat(k)) and vec(r)=(3hat(i)-hat(k))+s(4hat(j)-16hat(k)) intersect and find the position vector of their point of intersection.

Show that the lines vecr= (hat i+hat j+hat k)+t(hat i-hat j+hat k) and vecr= (3 hati-hatk)+s(4 hat j-16 hat k) intersect and find the position vector of their point of their point of intersection.

Find a unit vector perpendicular to both the vectors 2 hat (i) - 3 hat (j) + 6 hat (k) and 3 hat (j) - 4 hat (k) . Also find the sine of the angle between the given vectors .

Find the angle made by the vector sqrt(2) hat (i) + hat (j) + hat (k) with the y - axis

(i) Find the vector equation of a line passing through a point with position vector 2hat(i)-hat(j)+hat(k) , and parallel to the line joining the points -hat(i)+4hat(j)+hat(k) and hat(i)+2hat(j)+2hat(k) . Also find the cartesian equivalent of this equation. (ii) The cartesian equations of a line are 6x-2=3y+1=2z-2 . Find its direction ratios and also find vector equation of the line.