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3.4E: Exercises for Section 3.4

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    197420

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    Exercise \(\PageIndex{1}\)

    Find the area of the parallelogram formed by the vectors

    \(\vec{v}_1 = \langle 3, 1 \rangle, \quad \vec{v}_2 = \langle 2, -2 \rangle.\)

    Answer

    Area = \(8\)

    Exercise \(\PageIndex{2}\)

    For what values of \( k \) do the vectors \(\vec{v}_1 = \langle 2, -3 \rangle, \quad \vec{v}_2 = \langle -4, k \rangle \) form a parallelogram with zero area?

    Answer

    \(k = 6\)

    Exercise \(\PageIndex{3}\)

    Find the volume of the parallelepiped formed by \[ \vec v_1 = \langle 1, 2, 3 \rangle, \quad \vec v_2 = \langle 2, 3, 1 \rangle, \quad \vec v_3 = \langle3, 1, 2 \rangle. \]

    Answer

    Volume = \(18\)

    Exercise \(\PageIndex{4}\)

    For what values of \( k \) do the vectors \[ \vec v_1 = \langle 1, 2, 3\rangle, \quad \vec v_2 = \langle 4, k, 6\rangle, \quad \vec v_3 = \langle 7, 8, 9\rangle \] form a parallelepiped with zero volume?

    Answer

    \(k=6.\)

    Exercise \(\PageIndex{5}\)

    Find the area of the triangle with vertices \( A(0,0) \), \( B(2,3) \), and \( C(4,1) \).

    Answer

    Area = \(5\).

    Exercise \(\PageIndex{6}\)

    Find the area of the triangle with vertices \( (1,2) \), \( (4,6) \), and \( (5,3) \).

    Exercise \(\PageIndex{7}\)

    For what values of \( k \) do the vectors \[ (1,2,3), \quad (2,4,6), \quad (3,6,k) \] do not determine a 3 dimensional parallelepiped?

    Exercise \(\PageIndex{8}\)

    Suppose vectors \( \vec v_1, \vec v_2, \dots, \vec v_n\) are the rows of a skew symmetric matrix \(A\). Prove that the volume of the \(n\)-dimensional parallelepiped formed by \( \vec v_1, \vec v_2, \dots, \vec v_n\) has zero volume.

    Hint

    Since \(A\) is skew symmetric, \(A^T = -A\).

    Exercise \(\PageIndex{9}\)
    1. Show that if three vectors in \( \mathbb{R}^3 \) lie in the same plane, the determinant of their matrix representation is zero.
    2. Why does this happen geometrically?
    3. Provide an example of three linearly dependent vectors and compute their determinant.
    Exercise \(\PageIndex{10}\)
    1. Given three points \( A(x_1, y_1), B(x_2, y_2), C(x_3, y_3) \), show that they are collinear if and only if \[ \det \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix} = 0. \]
    2. Interpret this result in terms of area.
    3. What happens when four points in \( \mathbb{R}^3 \) satisfy a similar determinant condition?
    Exercise \(\PageIndex{11}\)

    Suppose we start with the identity matrix \(I_3\). Perform the following operations to obtain matrix \(A\):

    • \(2R_1+R_2->R_2\)
    • \(3R_2+R_3->R_3\)
    • \(R_2 <-> R_3\)
    • \(-4R_2+R_1->R_1\)
    1. What is \(A\)?
    2. What is \( \det(A)\)?

    This page titled 3.4E: Exercises for Section 3.4 is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Doli Bambhania, Fatemeh Yarahmadi, and Bill Wilson.