Решение математических задач по шагам онлайн

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Here are the step-by-step solutions for all four questions presented in the image:

Question 1

Problem: \(PQRS\) is a quadrilateral with \(P(4, 4)\), \(S(8, 8)\) and \(R(12, 8)\). If \(\vec{PQ} = 4\vec{SR}\), find the coordinates of \(Q\).

Step 1: Find the vector \(\vec{SR}\)

Given the coordinates of \(S(8, 8)\) and \(R(12, 8)\):
$\(\vec{SR} = R - S = \begin{pmatrix} 12 - 8 \\ 8 - 8 \end{pmatrix} = \begin{pmatrix} 4 \\ 0 \end{pmatrix}\)$

Step 2: Use the relation \(\vec{PQ} = 4\vec{SR}\)

Step 3: Find the coordinates of \(Q(x, y)\)

Since \(\vec{PQ} = Q - P\), we have:
$\(\vec{PQ} = \begin{pmatrix} x_Q - x_P \\ y_Q - y_P \end{pmatrix} = \begin{pmatrix} 16 \\ 0 \end{pmatrix}\)$

Given \(P(4, 4)\):
$\(x_Q - 4 = 16 \implies x_Q = 20\)$
$\(y_Q - 4 = 0 \implies y_Q = 4\)$

• Answer for Q1: \(Q = (20, 4)\)

Question 2

Problem: If \(\vec{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\), \(\vec{b} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}\) and \(\vec{r} = \vec{a} + \frac{1}{2}(\vec{a} - \vec{b})\), find \(|\vec{r}|\).

Step 1: Compute \(\vec{a} - \vec{b}\)

Step 2: Multiply by \(\frac{1}{2}\)

Step 3: Compute \(\vec{r}\)

Step 4: Find the magnitude \(|\vec{r}|\)

• Answer for Q2: \(|\vec{r}| = \sqrt{5}\) (or \(\approx 2.24\))

Question 3

Problem: The coordinates of the vertices of a parallelogram \(WXYZ\) are \(W(1, 6)\), \(X(2, 2)\), \(Y(5, 4)\) and \(Z(a, b)\). Find:
i. \(\vec{WX}\)
ii. \(\vec{ZY}\)
iii. Coordinate of \(Z\)

Part i: Find \(\vec{WX}\)

Part ii: Find \(\vec{ZY}\)

Since \(WXYZ\) is a parallelogram, opposite sides are equal and parallel: \(\vec{WX} = \vec{ZY}\).
Therefore:
$\(\vec{ZY} = \begin{pmatrix} 1 \\ -4 \end{pmatrix}\)$

Part iii: Find the coordinates of \(Z(a, b)\)

Using the vector \(\vec{ZY} = Y - Z\):
$\(\begin{pmatrix} 1 \\ -4 \end{pmatrix} = \begin{pmatrix} 5 - a \\ 4 - b \end{pmatrix}\)$

Set up the equations:
1. \(5 - a = 1 \implies a = 4\)
2. \(4 - b = -4 \implies b = 8\)

• Answers for Q3:
• i. \(\vec{WX} = \begin{pmatrix} 1 \\ -4 \end{pmatrix}\)
• ii. \(\vec{ZY} = \begin{pmatrix} 1 \\ -4 \end{pmatrix}\)
• iii. \(Z = (4, 8)\)

Question 4

Problem: The points \(M(2, 3)\), \(N(5, -2)\) and \(T(3, -5)\) are in the \(xy\)-plane. If \(k\vec{OM} + l\vec{ON} = \vec{MT}\), where \(k\) and \(l\) are scalar factors, find the values of \(k\) and \(l\). (Note: Assuming \(\vec{OM}, \vec{ON}, \vec{MT}\) represent the position vectors or displacements, let's use the coordinates directly).

(Assuming standard notation where vectors are formed from the origin: \(\vec{OM} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\), \(\vec{ON} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}\))

Step 1: Find vector \(\vec{MT}\)

Step 2: Set up the vector equation

This gives a system of two linear equations:
1. \(2k + 5l = 1\)
2. \(3k - 2l = -8\)

Step 3: Solve the system of equations

Multiply equation (1) by \(3\) and equation (2) by \(2\):
1. \(6k + 15l = 3\)
2. \(6k - 4l = -16\)

Subtract the second equation from the first:
$\((15l) - (-4l) = 3 - (-16)\)$
$\(19l = 19 \implies l = 1\)$

Substitute \(l = 1\) back into equation (1):
$\(2k + 5(1) = 1\)$
$\(2k + 5 = 1\)$
$\(2k = -4 \implies k = -2\)$

• Answer for Q4: \(k = -2\), \(l = 1\)