Use the figure to determine whether each statement is true...

Use the figure to determine whether each statement is true or false. Justify your answer. (a) $a=-d$ (b) $\mathbf{c}=\mathbf{s}$ (c) $\mathbf{a}+\mathbf{u}=\mathbf{c}$ (d) $\mathbf{v}+\mathbf{w}=-\mathbf{s}$ (e) $\mathbf{a}+\mathbf{w}=-2 \mathbf{d}$ (f) $\mathbf{a}+\mathbf{d}=0$ $(g) \mathbf{u}-\mathbf{v}=-2(\mathbf{b}+\mathbf{t})$ (h) $\mathbf{t}-\mathbf{w}=\mathbf{b}-\mathbf{a}$

Use the figure to determine whether each statement is true or false. Justify your answer.
(a) $a=-d$
(b) $\mathbf{c}=\mathbf{s}$
(c) $\mathbf{a}+\mathbf{u}=\mathbf{c}$
(d) $\mathbf{v}+\mathbf{w}=-\mathbf{s}$
(e) $\mathbf{a}+\mathbf{w}=-2 \mathbf{d}$
(f) $\mathbf{a}+\mathbf{d}=0$
$(g) \mathbf{u}-\mathbf{v}=-2(\mathbf{b}+\mathbf{t})$
(h) $\mathbf{t}-\mathbf{w}=\mathbf{b}-\mathbf{a}$

Key Concepts

Vector Subtraction and Negative Vectors

Vector subtraction is defined as the addition of the negative (or opposite) of a vector. Understanding negative vectors, which have the same magnitude as their positive counterparts but point in the opposite direction, is critical when interpreting expressions like u - v and a = -d. This concept enables the rearrangement of vector equations by considering subtraction as the addition of a negative vector.

Scalar Multiplication

Scalar multiplication involves scaling a vector by a real number (scalar), which affects the magnitude of the vector while preserving its direction (except in the case of a negative scalar, which also reverses the direction). This concept is important in statements where vectors are multiplied by constants, such as in -2(b + t) or -2d, allowing one to understand changes to vector lengths and directions that result from scalar factors.

Vector Diagram Interpretation

Interpreting vector diagrams involves reading and analyzing graphical representations of vectors to deduce relationships such as parallelism, direction, and magnitude. Diagrams help justify vector statements by visually confirming how vectors combine according to the head-to-tail rule, thereby supporting the verification of various vector equations provided in the problem.

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector. This operation is typically performed by aligning the tail of one vector with the head of another or using the parallelogram method. Understanding vector addition is crucial to interpreting sums like a + u = c or t - w = b - a by visualizing how these vectors add up in terms of magnitude and direction.

Vector Equality

Vector equality occurs when two vectors have exactly the same magnitude and direction. This concept is key when evaluating statements such as a = -d or c = s. It implies that even if the components of the vectors are written differently, their geometric representation in the vector space must be identical for the vectors to be considered equal.

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