Question

If p=4i-8k and Q=2i-mj+4k find m. If p and q have same direction

Answer 1

Answer:

solved

Explanation:

P=4i-8k and Q=2i-mj+4k

same direction means ∝ = 0

P.Q =(4i-8k) .(2i-mj+4k) = ║P║║Q║cos 0° = ║P║║Q║

8 - 0 -32 = √80√m²+20 = -24

m² + 20 = 24²/80 = 7.2

m² = - 12.8

other solution

multiples of j should be the same in both vectors

m = 0

Answer 2

Given:

Two vectors,$P=4i-j+8k$ and $Q=2i-mj+4k$

To find:

The value of m.

Solution:

We have been given that the two vectors P and Q lie in the same direction, hence, the angle between the vectors is zero.

If we calculate the scalar or dot product of the given vectors which says that.

$P$ . $Q=|P||Q|cos$θ

Since its is given that θ $=0$ and we know $cos(0)=1$

Therefore,

$P$ . $Q$ $=|P||Q|$

Now, the dot product of $P$ and $Q$ will be

$(4i-j+8k).(2i-mj+4k)=(\sqrt{(4)^{2}+ (-1)^{2}+ (8)^{2} })( \sqrt{(2)^{2}+ (-m)^{2}+ (4)^{2} })$

$8+m+32=(\sqrt{16+1+64} )(\sqrt{4+m^{2} +16 })$

$m+40=(\sqrt{81} )(\sqrt{m^{2} +20} )$

$(\frac{m+40}{9})^{2} = {(m^{2}+20 )$

$m^{2} +80m+1600=81m^{2} +1620$

$80m^{2} -80m+20=0$

$4m^{2} -4m+1=0$

$4m^{2} -2m-2m+1=0$

$2m(2m-1)-1(2m-1)=0$

$(2m-1)^{2} =0$

$m=\frac{1}{2}$

Final answer:

Hence, the value of m is $\frac{1}{2}$.

Although your question is incomplete, you might be referring to the question below.

If $P=4i-j+8k$ and $Q=2i-mj+4k$ . Find m. If $P$ and $Q$ have the same direction.