Magnitude of vector - seemingly impossible question!

[ajc43kpc]

I am helping someone with their National 5 Maths (in Scotland) and we are struggling with the question shown in the image. I also have a copy of the marking scheme and the answer they are looking for seems to be:

calculate the magnitude of V1, which they give as 5 (which is obviously the square root of 16 + 1 + 8), the squares of the coefficients of i, j and k
calculate the magnitude of V2, which they give as 10a (similarly)
and then it's just equating V1 and 2V2

but my question is WHY is the magnitude of V1 = 5? Doesn't it depend on the magnitudes of i, j and k? Am I being daft? There is no further information given about the vectors i, j and k - I even looked back at the previous question to see if it was a follow-on.

Please help!

Thanks.

 

[Otis]

… Doesn't it depend on the magnitudes of i, j and k? …

Yes, and those are unit vectors; their magnitude is 1. ?
\(\displaystyle \;\)

 

[ajc43kpc]

Yes, and those are unit vectors; their magnitude is 1. ?
\(\displaystyle \;\)

Thanks. I did wonder about that but could previously only find information about unit vectors being denoted by little hats. Your reply has helped me to find material that tells me this is an accepted convention for i, j and k (though it's not something which has been covered on the course.) The perils of using old past papers... Many thanks.

 

[pka]

Thanks. I did wonder about that but could previously only find information about unit vectors being denoted by little hats. Your reply has helped me to find material that tells me this is an accepted convention for i, j and k (though it's not something which has been covered on the course.) The perils of using old past papers... Many thanks.

You need to know how to find the norm of any vector.
If \(\displaystyle V=\alpha \bf{i}+\beta \bf{j}+\gamma\bf{k}\) then \(\displaystyle |V|=\sqrt{\alpha^2+\beta^2+\gamma^2}\)
In your case, \(\displaystyle a>0 \). WHY? So \(\displaystyle V_2=\sqrt{(8a)^2+(6a)^2}\).

 

[Dr.Peterson]

Thanks. I did wonder about that but could previously only find information about unit vectors being denoted by little hats. Your reply has helped me to find material that tells me this is an accepted convention for i, j and k (though it's not something which has been covered on the course.) The perils of using old past papers... Many thanks.

Yes, notation varies; i, j, k are common enough that we don't always use the full notation. See Wikipedia: https://en.wikipedia.org/wiki/Unit_vector#Orthogonal_coordinates:

They are often denoted using normal vector notation (e.g., i or

) rather than standard unit vector notation (e.g.,

). In most contexts it can be assumed that i, j, and k, (or

and

) are versors of a 3-D Cartesian coordinate system. The notations

,

,

, or

, with or without hat, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity (for instance with index symbols such as i, j, k, used to identify an element of a set or array or sequence of variables).