Question

A particle is moving with a constant speed; at time t=3 it has velocity <5,6> and acceleration . For what c is this possible?

Answer 1

if a particle is moving with a constant speed then its acceleration = 0
but i dont know the notation you used <5,6> and <c,-4>

Answer 2

HINT: That the speed is constant means that ||v(t)||=const or equivalently v(t)⋅v(t)=const. Differentiate both sides of this equality and use the properties of differentiation of vector functions (Stewart, page 566) and the properties of the dot product (Stewart, page 531) to obtain a relationship between the vectors v(t) and a(t)=v′(t). Use that relationship to find c.

Answer 3

v(t) * v(t) = const
d/dt [ v(t) * v(t) ] = d/dt [const]
2 * a(t) * v(t) = 0

it means that a(t) is perpendicular to v(t)

so
<5,6>* <c,−4> = 0
5c -24 = 0
c = 4.8