Q.Show that vectors A = 2i^ - 3j^ + 4k^ and B = -6i^ + 9j^ - 12k^ are parallel to each other.

lambda A (lambda negative) has magnitude |lambda||A| and points opposite to A.

When a vector A is multiplied by a real number lambda, the result lambda A is itself a vector.

• Its magnitude becomes |lambda| times the magnitude of A: |lambda A| = |lambda| x |A|.
• Its direction depends on the sign of lambda:
  • If lambda is positive, lambda A points in the SAME direction as A (just scaled in length).
  • If lambda is negative, lambda A points in the OPPOSITE direction to A (reversed by 180 degrees), and its magnitude is scaled by |lambda|. …

B works out to exactly −3 times A component-by-component, confirming the two vectors lie along the same line — they are parallel (specifically anti-parallel, pointing in opposite senses).

Step 1: Write down the given vectors.

A = 2i^ − 3j^ + 4k^

B = −6i^ + 9j^ − 12k^

Step 2: Test whether B is a scalar multiple of A.

Compare each component's ratio:

• i^ component: −6/2 = −3
• j^ component: 9/(−3) = −3
• k^ component: −12/4 = −3

Step 3: Confirm the ratio is the same, λ = −3, for every component.

B = −3(2i^ − 3j^ + 4k^) = −3A

Step 4: Interpret the result. …