Vector Calculus Answers

a=2i-3j-k, b=i+4j+3k then what;s their angle? if i use dot product formula it comes arc cos[(-sqrt(13)/{(2*sqrt(7)}]

and if i use cross product formula then arc sin[(sqrt(15/28)] . two angles are not same if i convert to degree , then 1st one comes 132.951978120924 and the 2nd one 47.048021879076. it become equal that time if (180-47.048021879076) . because sin(180-x)=sin x . but for inverse function , we use the least value. like arc sin(1/2)= 30 degree, not 150 degree. so if i think at this angle the angle can't be equal. so which one is correct?

a vector = 2i+3j-k , find a unit vector parallel to this vector ? should i use (±)? unit vector = ±( 2i+3j-k)/sqrt(14).?

i m confused to use (±). if (±) needed , then why ?? someone says opposite direction that's why .please show me the two directions for using (±) in diagram

a=2i+3j-k , b=6i-2j+5k find a unit vector parallel to the resultant of these vectors? here should i use (±) to find unit vector?
if needed then why? i can't figure out this. please show me in diagram, where it is +ve and where it is -ve

∇(x/r^3)=? ∇(r vector/r^3)=?
where
r vector = xi+yj+zk

∇(1/r)=? ∇^2 (1/r)=?
where
r vector = xi+yj+zk

A hiker travels 2.5km due North followed by 4.5km on a bearing 132°. Calculate the distance and bearing of the hiker's final position from his initial position.

In an orienteering race the first checkpoint is 500m from the start and on a bearing of 030°. The second checkpoint is 400m from the first checkpoint and 600m from the start. Find, to the nearest degree, the two possible bearings of checkpoint 2 from the start.

In a cricket match a batsman hits the ball and a fielder, 30 m away, fails to stop the ball but deflects it and slows it down. The ball comes to rest 20m from where the fielder deflected it. Find the distance from where the ball was hit to where it came to rest. ( Angle between 30m and 20m is 110° )

Two forces have magnitudes of 10N and 15N and the angle between them is 45°. Find the magnitude of the resultant and the angle it makes with the smaller of the two forces.

Two forces have magnitudes of 8N and 12 N and the angle between them is 130°. Find the magnitude of the resultant and the angle it makes with the larger of the two forces.