We read your photo as:

Given that p = [-2, 3], q = [2, 3] and r = [1, -5], calculate the values of m and n from the equation r = mp + nq, where m and n are scalars.

Mathematics

ScanSolve AI Answer

Answer

m=1,n=−1m=1, n=-1

Step-by-step solution

6 steps

  1. Substitute the given vectors into the equation r=mp+nqr = mp + nq.

    (1−5)=m(3−2)+n(23)\begin{pmatrix} 1 \\ -5 \end{pmatrix} = m \begin{pmatrix} 3 \\ -2 \end{pmatrix} + n \begin{pmatrix} 2 \\ 3 \end{pmatrix}

  2. Perform scalar multiplication and vector addition.

    (1−5)=(3m−2m)+(2n3n)\begin{pmatrix} 1 \\ -5 \end{pmatrix} = \begin{pmatrix} 3m \\ -2m \end{pmatrix} + \begin{pmatrix} 2n \\ 3n \end{pmatrix} (1−5)=(3m+2n−2m+3n)\begin{pmatrix} 1 \\ -5 \end{pmatrix} = \begin{pmatrix} 3m + 2n \\ -2m + 3n \end{pmatrix}

  3. Form a system of two linear equations by equating the corresponding components.

    1=3m+2n(Equation 1)1 = 3m + 2n \quad \text{(Equation 1)} −5=−2m+3n(Equation 2)-5 = -2m + 3n \quad \text{(Equation 2)}

  4. Solve the system of equations.

    Multiply Equation 1 by 2 and Equation 2 by 3 to eliminate mm.

    2×(3m+2n=1)  ⟹  6m+4n=2(Equation 3)2 \times (3m + 2n = 1) \implies 6m + 4n = 2 \quad \text{(Equation 3)} 3×(−2m+3n=−5)  ⟹  −6m+9n=−15(Equation 4)3 \times (-2m + 3n = -5) \implies -6m + 9n = -15 \quad \text{(Equation 4)}

  5. Add Equation 3 and Equation 4.

    (6m+4n)+(−6m+9n)=2+(−15)(6m + 4n) + (-6m + 9n) = 2 + (-15) 13n=−1313n = -13 n=−1313n = \frac{-13}{13} n=−1n = -1

  6. Substitute the value of nn into Equation 1 to find mm.

    3m+2(−1)=13m + 2(-1) = 1 3m−2=13m - 2 = 1 3m=1+23m = 1 + 2 3m=33m = 3 m=33m = \frac{3}{3} m=1m = 1

    The values are m=1m=1 and n=−1n=-1.

    The final answer is m=1,n=−1m=1, n=-1.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Mathematics Questions

QTwo sides of a triangle are represented by the vectors m=(2/3) and n=(3/2). If r=pm+qn where…QThe coordinate of P, Q and R are (2, m), (−3, 1) and (6, n) respectively. If the length of PQ is…QGiven p = 5 −7 , q = −1 3 , and r = 2 1 .QGiven vectors p = 2 −2 and q = 3 4 , solve for vector r in the equation (1)/(2)p - q + r = 0 0 .QGiven 2p − 3q + r = 0, p = 2i − 7j, and q = −5i − 9j, find the magnitude of vector r.
See all Mathematics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.