Answer the following as true or false.
a⃗ and -a⃗ are collinear.
True. Vectors a⃗ and -a⃗ are parallel to the same line (one is a scalar multiple of the other, with scalar -1), hence they are collinear.
Two collinear vectors are always equal in magnitude.
False. Two collinear vectors are related by a scalar, but the scalar may not have a magnitude of 1; for example, a⃗ and 2a⃗ are collinear but have different magnitudes.
Two vectors having same magnitude are collinear.
False. Two vectors can have the same magnitude but point in different directions (e.g., perpendicular vectors), so they are not necessarily collinear.
Two collinear vectors having the same magnitude are equal.
False. Two collinear vectors having the same magnitude can have opposite directions (e.g., a⃗ and -a⃗); equal vectors must have both the same magnitude and the same direction.
Explanation
These answers are derived from the properties of collinear and equal vectors provided in the context. Question 19 indicates that collinear vectors are related by a scalar (b⃗ = λa⃗). This supports statement (i) as True (λ = -1). It supports statement (ii) as False because |λ| can be any value, not just 1. Statement (iii) is False because vectors can have the same magnitude but different directions (e.g., perpendicular vectors mentioned in Q11). Statement (iv) is False because collinear vectors with the same magnitude can point in opposite directions (λ = -1), whereas equal vectors must have the same direction as well, as implied by the distinction between collinear and equal vectors in Example 3.