Question

If \[ \vec{a}=4\vec{i}+6\vec{j},\quad \vec{b}=3\vec{j}+4\vec{k} \] and \(\vec{c}\) is the projection vector of \(\vec{a}\) on \(\vec{b}\), then \(\vec{c}\) and \(|\vec{c}|\) respectively are:

• A. \(\dfrac{18}{25}\vec{b},\ \dfrac{18}{5}\)
• B. \(\dfrac{18}{5}\vec{b},\ 18\)
• C. \(\dfrac{25}{18}\vec{b},\ \dfrac{18}{5}\)
• D. \(\dfrac{5}{18}\vec{b},\ \dfrac{5}{18}\)

Hint:

The projection vector of \(\vec{a}\) on \(\vec{b}\) is \(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\vec{b}\), while its magnitude is \(\frac{|\vec{a}\cdot\vec{b}|}{|\vec{b}|}\).

Answer 1

The Correct Option is A

Step 1: State the vector projection formula.
The projection of a⃗ onto b⃗ is c⃗ = [(a⃗·b⃗)/|b⃗|²] b⃗.

Step 2: Compute the dot product a⃗·b⃗.
For a⃗ = 4î + 6ĵ and b⃗ = 3ĵ + 4k̂: a⃗·b⃗ = (4)(0) + (6)(3) + (0)(4) = 18.

Step 3: Compute the squared magnitude of b⃗.
|b⃗|² = 0² + 3² + 4² = 9 + 16 = 25, so |b⃗| = 5.

Step 4: Form the projection vector.
c⃗ = (18/25) b⃗.

Step 5: Determine the magnitude of the projection.
|c⃗| = (18/25)|b⃗| = (18/25) × 5 = 18/5.

Step 6: Final conclusion.
The projection vector is (18/25)b⃗ with magnitude 18/5.