Question

a,b and c are three orthogonal vectors with magnitude 3, 4 and 12 respectively. What will be the ratio of magnitude of their sum and sum of their magnitude?

Answer 1

For this problem, we have the freedom to pick the vectors such that each one  is in the standard $x$ direction, another in the standard $y$ direction and the last one in the standard $z$ direction as shown below,

$\bar{a}=3i$, $\bar{b}=4j$ and $\bar{c}=12k.$

The sum $\bar{S}$ of these vectors is then,

$\bar{S} =\bar{a}+\bar{b}+\bar{c}=3i+4j+12k$

The magnitude of $\bar{S}$ is calculated as follows,

$|S|=\sqrt{3^2+4^2+12^2}=\sqrt{169}=13.$

The sum of the magnitudes of the vectors $Z$ is,

$Z=3+4+12=19.$

The ratio of these terms, $\frac{S}{Z}=\frac{13}{19}$

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

ᴀ,ʙ ᴀɴᴅ ᴄ ᴀʀᴇ ᴛʜʀᴇᴇ ᴏʀᴛʜᴏɢᴏɴᴀʟ ᴠᴇᴄᴛᴏʀꜱ ᴡɪᴛʜ ᴍᴀɢɴɪᴛᴜᴅᴇ 3, 4 ᴀɴᴅ 12 ʀᴇꜱᴘᴇᴄᴛɪᴠᴇʟʏ. ᴡʜᴀᴛ ᴡɪʟʟ ʙᴇ ᴛʜᴇ ʀᴀᴛɪᴏ ᴏꜰ ᴍᴀɢɴɪᴛᴜᴅᴇ ᴏꜰ ᴛʜᴇɪʀ ꜱᴜᴍ ᴀɴᴅ ꜱᴜᴍ ᴏꜰ ᴛʜᴇɪʀ ᴍᴀɢɴɪᴛᴜᴅᴇ?

♣ ᴀɴꜱᴡᴇʀ :

ᴀ = 3ɪ

ʙ = 4ᴊ

ᴄ = 12ᴋ

ꜱ = ᴀ - ʙ + ᴄ

  = 3ɪ - 4ᴊ + 12ᴋ

ᴢ = 3 - 4 + 12

  = 3 + 12 - 4

  = 15 - 4

  = 11