PUCET (Punjab) Linear Combination — practice questions

Explore 2 free Linear Combination questions in PUCET (Punjab) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Linear Combination questions and exercises with answers

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Find the value of $\lambda$ and $\mu$ if the non-zero, non-collinear vectors $\vec a$ and $\vec b$ satisfy $\vec a + 3\vec b = 2\lambda \vec a - \mu \vec b$:Read the answer and worked solution →Vector $4\hat i + 13\hat j - 18\hat k$ expressed as a linear combination of $\hat i - 2\hat j + 3\hat k$ and $2\hat i + 3\hat j - 4\hat k$ has coefficients $(m, n)$:Read the answer and worked solution →

Worked practice questions

Find the value of $\lambda$ and $\mu$ if the non-zero, non-collinear vectors $\vec a$ and $\vec b$ satisfy $\vec a + 3\vec b = 2\lambda \vec a - \mu \vec b$:

$\lambda = 1/2,\ \mu = -3$
$\lambda = -1/2,\ \mu = 3$
$\lambda = 2,\ \mu = -3$
$\lambda = 1,\ \mu = -3$

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Correct answer: A. $\lambda = 1/2,\ \mu = -3$

Non-collinear $\vec a, \vec b$ form a basis ⇒ coefficients on each side must match. Coefficient of $\vec a$: $1 = 2\lambda$ ⇒ $\lambda = 1/2$. Coefficient of $\vec b$: $3 = -\mu$ ⇒ $\mu = -3$.

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Vector $4\hat i + 13\hat j - 18\hat k$ expressed as a linear combination of $\hat i - 2\hat j + 3\hat k$ and $2\hat i + 3\hat j - 4\hat k$ has coefficients $(m, n)$:

$(2, 3)$
$(-2, 3)$
$(3, -2)$
$(-3, 2)$

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Correct answer: B. $(-2, 3)$

Set $4\hat i + 13\hat j - 18\hat k = m(\hat i - 2\hat j + 3\hat k) + n(2\hat i + 3\hat j - 4\hat k)$. **Equating components:** $m + 2n = 4$ and $-2m + 3n = 13$. **Solving:** $m = -2, n = 3$. Check the $\hat k$ component: $3(-2) - 4(3) = -18$ ✓.