Determine whether the following statements are true or false. If it is true, exp

Question

Determine whether the following statements are true or false. If it is true, explain why. If it is false, give a counter example. (a) (T/F) For any vectors u vector and upsilon vector, we have u vector middot upsilon vector = upsilon vector middot u vector. (b) (T/F) For any vectors u vector and upsilon vector, we have u vector times upsilon vector = upsilon vector times u vector. (c) (T/F) For any vectors u vector and upsilon vector, we have (u vector times upsilon vector) middot u vector = 0. (d) (T/F) If u vector middot upsilon vector = 0, then u vector = 0 vector or upsilon vector = 0 vector. (e) (T/F) If u vector times upsilon vector = 0 vector, then u vector = 0 vector or upsilon vector = 0 vector.

Explanation / Answer

(a) u.v = v.u is true for any vectors u and v because dot product is a scalar quantity and it will be same weather u.v or v.u

(b) uxv = vxu is false for any vectors u and v because cross product is a vector quantity which has direction and it will be chnged for uxv and vxu

example: Let vectors u = i , v = j

uxv = ixj => k

vxu =jxi => -k

here both uxv and vxu are not equal. Thus we have proved counter example.

(c) (uxv).u = 0 is true statement because (uxv).u is always zero.

(d) if u.v=0 then u = 0 or v = 0 is false statement because u.v can also be zero if u and v vectors are perpendicular.

example:

example: Let non zero vectors u = i , v = -j

If we calculate u.v then it will be 0. Thus we have proved counter example.

(e) if uxv=0 then u = 0 or v = 0 is false statement because uxv can also be zero if u and v vectors are parallel.

example: Let non zero vectors u = i , v = i

If we calculate uxv then it will be 0. Thus we have proved counter example.

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